Introduction
Operator learning seeks a single approximation of the map from the data of an equation to its solution, enabling evaluation at new inputs without solving each instance input. The operator approximation theorems in [9 ] , DeepONet [31 ] , and Fourier neural operators [27 , 20 ] have established such approximation for maps between function spaces. For stochastic partial differential equations (SPDEs), we study solution operators that map initial data and random forcing to the corresponding solutions. The central challenge is that the forcing may be distribution-valued, requiring an operator approximation framework that accommodates inputs without pointwise values.
This paper develops a finite-measurement approximation theory for such SPDE solution operators. Our starting point is the probabilistic operator framework in [5 ] , which combines integrability, tail estimates, and Lipschitz continuity for models driven by finite-dimensional stochastic processes. Extending this approach to distribution-valued space–time forcing requires an encoding that is well defined on the noise space and an estimate that propagates its error through the equation. We use Fourier measurements to construct the encoding and Schauder stability to control the resulting error. The resulting branch–trunk network approximates the joint dependence on the initial condition and the forcing in stochastic moments of the uniform space–time norm, with explicit measurement and reconstruction rates.
We consider the semilinear stochastic heat equation
∂ t u − Δ u = F ( u ) + η , ( t , x ) ∈ ( 0 , T ] × T d , u ( 0 , x ) = u 0 ( x ) , x ∈ T d . (1) \begin{aligned}
\partial_t u-\Delta u &= F(u)+\eta,\qquad(t,x)\in(0,T]\times\mathbb{T}^{d},\\
u(0,x) &= u_0(x),\qquad x\in\mathbb{T}^{d}. \tag*{(1)}
\end{aligned} ∂ t u − Δ u u ( 0 , x ) = F ( u ) + η , ( t , x ) ∈ ( 0 , T ] × T d , = u 0 ( x ) , x ∈ T d . (1) 2020 Mathematics Subject Classification. 60H15, 68T07, 35R60, 41A25, 60H35, 35Q30.
Key words and phrases. Deep operator networks, neural operators, universal approximation, semilinear stochastic partial differential equations, Gaussian forcing, parabolic Hölder–Besov spaces, Fourier measurements, Wiener chaos, stochastic Navier–Stokes equations.
QF is partially supported by the National Science Foundation under grant #DMS-2420029. WW was partially supported by the Simons Foundation Travel Support for Mathematicians (No. 0007730).
where d ≥ 2 d \ge2 d ≥ 2 F : R → R F : \mathbb{R} \to\mathbb{R} F : R → R u 0 ∈ C ( T d ) u_0 \in C(\mathbb{T}^d) u 0 ∈ C ( T d ) η \eta η
T θ ( u 0 , η ) ( t , x ) = ∑ ℓ = 1 N 1 a ℓ ( u ⃗ 0 , c J ( η ) ) q ℓ ( t , x ) , \mathcal{T}_\theta(u_0,\eta)(t,x)=\sum_{\ell=1}^{N_1}a_\ell(\vec{u}_0,c_J(\eta))q_\ell(t,x), T θ ( u 0 , η ) ( t , x ) = ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( η )) q ℓ ( t , x ) , where u ⃗ 0 \vec{u}_0 u 0 N 0 N_0 N 0 c J ( η ) c_J(\eta) c J ( η ) N J N_J N J q ∈ [ 1 , ∞ ) q \in[1,\infty) q ∈ [ 1 , ∞ ) ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 )
sup u 0 ∈ G 0 ( E [ ∥ S ( u 0 , η ) − T θ ( u 0 , η ) ∥ C ( D T ) q ] ) 1 / q ≤ ε . \sup_{u_0\in\mathcal{G}_0}\left(\mathbb{E}\left[\left\|S(u_0,\eta)-\mathcal{T}_\theta(u_0,\eta)\right\|_{C(D_T)}^q\right]\right)^{1/q}\le\varepsilon. u 0 ∈ G 0 sup ( E [ ∥ S ( u 0 , η ) − T θ ( u 0 , η ) ∥ C ( D T ) q ] ) 1/ q ≤ ε . Thus the approximation controls full solution paths, uniformly over the initial-data class, while retaining the prescribed coupling between the noise input and the solution output. The main analytic step is a quantitative reduction of the rough/noisy input. A smooth finite-rank Fourier cutoff satisfies
∥ ( I − Π J ) η ∥ C β 1 ≲ 2 − J δ ∥ η ∥ C β 0 . \left\|(I-\Pi_J)\eta\right\|_{C^{\beta_1}}\lesssim2^{-J\delta}\|\eta\|_{C^{\beta_0}}. ∥ ( I − Π J ) η ∥ C β 1 ≲ 2 − J δ ∥ η ∥ C β 0 . The main difficulty is to turn this finite-dimensional reduction into an approximation of the SPDE solution operator. The forcing may not have pointwise values, and its truncation error must be measured in a topology in which the solution map is stable. We combine the stability estimate for (1) with partition-of-unity approximation of the finite-dimensional input dependence and parabolic interpolation of the output. This extends the function, functional, and operator approximation approach in [28 , 5 ] to distribution-valued forcing, with explicit bounds on the numbers of input measurements and trunk functions.
Another difficulty is that the Gaussian input is not confined to a compact set. Therefore Uniform approximation on bounded sets alone does not yield the stochastic moment estimate above. We control the growth of the neural approximation outside these sets by a coordinatewise projection, and use Fernique’s theorem to estimate the remaining Gaussian tail. The construction consequently gives approximation in every fixed finite stochastic moment without imposing compact support on the noise law.
For stochastic incompressible Navier–Stokes equations, the quadratic transport term contains a spatial derivative, and the solution map is only locally Lipschitz. We establish a common deterministic existence interval for the admissible initial velocities and forcings in bounded sets, together with their Fourier truncations. On this interval, which is independent of the approximation accuracy, local stability yields uniform operator approximation on an event of high probability. The measurement bounds distinguish the regularity required of the initial velocity from that of the forcing.
Related literature
The deterministic operator-learning literature supplies the approximation architecture used below. Alongside DeepONet, the Fourier neural operator in [27 ] and the general neural-operator framework in [20 ] provide resolution-independent parameterizations of maps between function spaces. Universal approximation and error estimates for Fourier neural operators are proved in [19 ] ; the encoding–approximation–reconstruction analysis of DeepONets in [23 ] is particularly close to the error decomposition used here. Random-feature operator learning [35 ] gives a probabilistic alternative, and its statistical error analysis extends to infinite-dimensional, vector-valued regression [24 ] . Learning linear operators from noisy input–output pairs is studied in [12 ] . These general results also show why quantitative branch-network bounds are difficult without additional structure: a generic Lipschitz operator on a high-dimensional encoded input class retains the curse of dimensionality.
Operator learning has also been extended to geometry-dependent PDE solution maps. The DIMON framework [44 ] transports problems on a family of diffeomorphic domains to a common reference domain, where the dependence on the geometry and initial or boundary data is learned.
Finite-dimensional inputs and outputs also arise only when parameters or observables of an underlying PDE solution are available. Fourier neural mappings for such parameter-to-observable maps are developed in [18 ] . In the inverse direction, neural inverse operators combine DeepONet and Fourier-neural-operator components to learn PDE inverse maps [34 ] ; see [36 ] for a broader probabilistic account of operator learning for inverse problems. Those works concern statistical recovery from observations, whereas the present paper constructs a pathwise approximation of a forward SPDE solution map after encoding its distribution-valued forcing.
Operator learning for stochastic models has applications in mathematical finance, control, and game theory. Solution operators for structured families of forward–backward stochastic differential equations with random terminal time are approximated in [13 ] . In dynamic Stackelberg games, attention-based neural operators approximate the follower’s best-response operator [1 ] . Neural operators for families of convex optimization problems are developed in [21 ] , with applications to stochastic optimal control, portfolio optimization, and hedging. These operator formulations complement methods such as the Deep-BSDE method [16 ] , which approximates the solution of a fixed equation. For optimal execution problems with unknown price impact, the authors of [32 ] combine in-context learning of the price-impact operator from observed trade and impact trajectories with neural-network optimization of the execution policy. Another interesting in-context operator learning on spaces of probability measures is studied in [11 ] , using samples from source and target distributions to approximate optimal transport maps.
The authors of [22 ] study the recovery of a discrete-time linear transition operator from sparse observations of its powers. In [30 ] , the authors establish consistency and learning rates for interaction kernels inferred from multiple trajectories of stochastic particle systems. These works concern inference of the governing dynamics, whereas one can fix the SDE and approximate its solution operator as a function of the initial condition and distribution-valued forcing. In [3 ] , the authors apply separate adaptive optimizers to the individual loss terms before combining their updates. Curvature-aware corrections to first-order optimizers are developed in [2 ] , while a regularized quasi-Newton method for high-accuracy training is proposed in [41 ] . These works address training algorithms; our results concern the existence and size of approximating networks.
Several architectures have been designed specifically for stochastic PDEs. Neural SPDEs [38 ] build the mild-solution structure into a resolution-invariant model. Feature engineering with regularity structures [10 ] , NORS [17 ] , and the deep latent regularity network [14 ] use polynomial or renormalized features of the driving noise. Such features are indispensable for singular equations in which the solution and its nonlinearities are not classically defined. SPDEBench [26 ] provides a numerical comparison across regular and singular stochastic equations. The present paper addresses a different question: for additive Gaussian forcing whose heat convolution is continuous, are linear measurements of the distributional input already sufficient for a pathwise operator-approximation theorem? The answer is affirmative under Assumption 2.2.
Navier–Stokes solution operators have also been used as important test cases for neural operators. Quantitative approximation results for deterministic incompressible Navier–Stokes appear in [19 ] , while Neural SPDEs [38 ] and NORS [17 ] include stochastic Navier–Stokes experiments. In [43 ] , the authors introduced a conformal prediction approach from statistics to study the operator learning of deterministic Navier–Stokes equations. Our result is complementary to these works: We prove that finitely many linear measurements of the forcing and initial velocity suffice for uniform approximation on a common local existence interval, on an event of high probability. The measurement counts in Theorem 5.8 separate the forcing regularity from the initial velocity regularity.
The finite measurements also connect the construction to Wiener chaos. Classical chaos solutions of stochastic evolution equations are developed in [33 , 29 ] ; recent work learns the deterministic propagators in truncated chaos expansions [37 , 39 , 40 ] . The precise relation is recorded in Lemma 3.11: for the linear equation, the finite-rank solution used here is exactly the truncation of the first-chaos expansion to the retained coordinates. For a general globally Lipschitz nonlinearity, the present proof instead uses Schauder stability and does not require a closed propagator hierarchy. Our approximation argument follows the function, functional, and operator approximation results of [5 ] . Deep-BSDE methods target the solution of a fixed equation, whereas the present branch–trunk network approximates the joint dependence of the solution on the initial condition and the forcing realization.
The paper is organized as follows. Section 2 states preliminaries, the analytic assumptions, and proves well-posedness and Lipschitz stability. Section 3 constructs the finite-dimensional measurement map and the deep operator architecture. Section 4 proves the function, functional, and operator approximation results. Section 5 develops the high-probability local extension to stochastic Navier–Stokes equations.
Preliminaries
In this section, we specify the function spaces and give some preliminaries.
Fourier multipliers and function spaces
Fix T > 0 T>0 T > 0
T d = ( R / 2 π Z ) d , T T = R / T Z , Λ T = T T × T d , D T = [ 0 , T ] × T d . \mathbb{T}^{d}=(\mathbb{R}/2\pi\mathbb{Z})^{d},\qquad\mathbb{T}_{T}=\mathbb{R}/T\mathbb{Z},\qquad\Lambda_{T}=\mathbb{T}_{T}\times\mathbb{T}^{d},\qquad D_{T}=[0,T]\times\mathbb{T}^{d}. T d = ( R /2 π Z ) d , T T = R / T Z , Λ T = T T × T d , D T = [ 0 , T ] × T d . The forcing is realized on the space–time torus Λ T \Lambda_{T} Λ T D T D_{T} D T D T D_{T} D T
d s ( ( t , x ) , ( s , y ) ) : = ∣ t − s ∣ 1 / 2 + d T d ( x , y ) , d T d ( x , y ) : = min m ∈ Z d ∣ x − y − 2 π m ∣ . d_{\mathfrak{s}}((t,x),(s,y)):=|t-s|^{1/2}+d_{\mathbb{T}^{d}}(x,y),\qquad d_{\mathbb{T}^{d}}(x,y):=\min_{m\in\mathbb{Z}^{d}}|x-y-2\pi m|. d s (( t , x ) , ( s , y )) := ∣ t − s ∣ 1/2 + d T d ( x , y ) , d T d ( x , y ) := m ∈ Z d min ∣ x − y − 2 π m ∣. This distance corresponds to the scaling s = ( 2 , 1 , … , 1 ) \mathfrak{s}=(2,1,\ldots,1) s = ( 2 , 1 , … , 1 ) ∣ s ∣ = d + 2 |\mathfrak{s}|=d+2 ∣ s ∣ = d + 2 0 < γ < 1 0<\gamma<1 0 < γ < 1
∥ f ∥ C s γ ( D T ) : = ∥ f ∥ ∞ + sup z ≠ z ′ ∣ f ( z ) − f ( z ′ ) ∣ d s ( z , z ′ ) γ . \|f\|_{C_{\mathfrak{s}}^{\gamma}(D_{T})}:=\|f\|_{\infty}+\sup_{z\ne z'}\frac{|f(z)-f(z')|}{d_{\mathfrak{s}}(z,z')^{\gamma}}. ∥ f ∥ C s γ ( D T ) := ∥ f ∥ ∞ + z = z ′ sup d s ( z , z ′ ) γ ∣ f ( z ) − f ( z ′ ) ∣ . All function-space norms below are deterministic norms. Expectations are taken separately when these norms are applied to random functions. For vector-valued functions, we take the maximum over components.
The orthonormal Fourier basis on Λ T \Lambda_{T} Λ T
ε n , k ( t , x ) : = ( T ( 2 π ) d ) − 1 / 2 e 2 π i n t / T + i k ⋅ x , ( n , k ) ∈ Z × Z d . \varepsilon_{n,k}(t,x):=(T(2\pi)^{d})^{-1/2}e^{2\pi int/T+ik\cdot x},\qquad(n,k)\in\mathbb{Z}\times\mathbb{Z}^{d}. ε n , k ( t , x ) := ( T ( 2 π ) d ) − 1/2 e 2 π in t / T + ik ⋅ x , ( n , k ) ∈ Z × Z d . For a distribution f f f f ^ ( n , k ) \widehat{f}(n,k) f ( n , k ) ε n , k ‾ \overline{\varepsilon_{n,k}} ε n , k f ^ ( n , k ) = ⟨ f , ε n , k ‾ ⟩ \widehat{f}(n,k)=\langle f,\overline{\varepsilon_{n,k}}\rangle f ( n , k ) = ⟨ f , ε n , k ⟩ e k ( x ) = ( 2 π ) − d / 2 e i k ⋅ x e_{k}(x)=(2\pi)^{-d/2}e^{ik\cdot x} e k ( x ) = ( 2 π ) − d /2 e ik ⋅ x
Now we fix a smooth dyadic decomposition compatible with parabolic scaling: choose a nonincreasing function ζ ∈ C ∞ ( [ 0 , ∞ ) ; [ 0 , 1 ] ) \zeta\in C^{\infty}([0,\infty);[0,1]) ζ ∈ C ∞ ([ 0 , ∞ ) ; [ 0 , 1 ]) [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 4 / 3 , ∞ ) [4/3,\infty) [ 4/3 , ∞ )
ρ ( ω , κ ) : = ( ω 2 + ∣ κ ∣ 4 ) 1 / 4 , χ ( ω , κ ) : = ζ ( ρ ( ω , κ ) ) . \rho(\omega,\kappa):=(\omega^{2}+|\kappa|^{4})^{1/4},\qquad\chi(\omega,\kappa):=\zeta(\rho(\omega,\kappa)). ρ ( ω , κ ) := ( ω 2 + ∣ κ ∣ 4 ) 1/4 , χ ( ω , κ ) := ζ ( ρ ( ω , κ )) . Although ρ \rho ρ χ \chi χ ρ ( λ 2 ω , λ κ ) = λ ρ ( ω , κ ) \rho(\lambda^{2}\omega,\lambda\kappa)=\lambda\rho(\omega,\kappa) ρ ( λ 2 ω , λκ ) = λ ρ ( ω , κ ) ρ ( ω , κ ) ≍ ∣ ω ∣ 1 / 2 + ∣ κ ∣ \rho(\omega,\kappa)\asymp|\omega|^{1/2}+|\kappa| ρ ( ω , κ ) ≍ ∣ ω ∣ 1/2 + ∣ κ ∣ δ ( ω , κ ) = ( λ 2 ω , λ κ ) \delta(\omega,\kappa)=(\lambda^{2}\omega,\lambda\kappa) δ ( ω , κ ) = ( λ 2 ω , λκ )
χ − 1 : = χ , χ j : = χ ∘ δ 2 j − 1 − χ ∘ δ 2 j , j ≥ 0. \chi_{-1}:=\chi,\qquad\chi_j:=\chi\circ\delta_{2^{j-1}}-\chi\circ\delta_{2^j},\qquad j\ge0. χ − 1 := χ , χ j := χ ∘ δ 2 j − 1 − χ ∘ δ 2 j , j ≥ 0. These symbols form a partition of unity in frequency. For j ≥ 0 j \ge0 j ≥ 0 χ j \chi_j χ j 2 j ≤ ϱ ≤ 2 j + 3 / 3 2^j \le\varrho\le2^{j+3}/3 2 j ≤ ϱ ≤ 2 j + 3 /3
Δ j s f : = ∑ n , k χ j ( n , k ) f ^ ( n , k ) ε n , k , \Delta_j^s f := \sum_{n,k} \chi_j(n,k)\widehat{f}(n,k)\varepsilon_{n,k}, Δ j s f := n , k ∑ χ j ( n , k ) f ( n , k ) ε n , k , Π J : = ∑ j = − 1 J Δ j s , m J ( n , k ) : = ς ( 2 − ( J + 1 ) ϱ ( n , k ) ) . \Pi_J := \sum_{j=-1}^{J} \Delta_j^s,\qquad m_J(n,k) := \varsigma\left(2^{-(J+1)}\varrho(n,k)\right). Π J := j = − 1 ∑ J Δ j s , m J ( n , k ) := ς ( 2 − ( J + 1 ) ϱ ( n , k ) ) . Thus Π J \Pi_J Π J m J m_J m J 0 ≤ m J ≤ m J + 1 ≤ 1 0 \le m_J \le m_{J+1} \le1 0 ≤ m J ≤ m J + 1 ≤ 1 m J → 1 m_J \to1 m J → 1 2 2 J 2^{2J} 2 2 J 2 J 2^J 2 J
Definition 2.1 (Parabolic Hölder–Besov spaces). For β ∈ R \beta\in\mathbb{R} β ∈ R
C β ( Λ T ) : = B ∞ , ∞ β , s ( Λ T ) \mathcal{C}^{\beta}(\Lambda_T) := B_{\infty,\infty}^{\beta,s}(\Lambda_T) C β ( Λ T ) := B ∞ , ∞ β , s ( Λ T ) be the inhomogeneous Besov space associated with the Littlewood–Paley decomposition ( Δ j s ) j ≥ − 1 (\Delta_j^s)_{j\ge-1} ( Δ j s ) j ≥ − 1
∥ f ∥ C β ( Λ T ) : = sup j ≥ − 1 2 j β ∥ Δ j s f ∥ L ∞ ( Λ T ) . \|f\|_{\mathcal{C}^{\beta}(\Lambda_T)} := \sup_{j\ge-1} 2^{j\beta}\|\Delta_j^s f\|_{L^\infty(\Lambda_T)}. ∥ f ∥ C β ( Λ T ) := j ≥ − 1 sup 2 j β ∥ Δ j s f ∥ L ∞ ( Λ T ) . We denote by C g β ( Λ T ) \mathcal{C}_g^\beta(\Lambda_T) C g β ( Λ T ) C ∞ ( Λ T ) C^\infty(\Lambda_T) C ∞ ( Λ T ) R × T d \mathbb{R} \times\mathbb{T}^d R × T d C β ( R × T d ) \mathcal{C}^\beta(\mathbb{R} \times\mathbb{T}^d) C β ( R × T d ) f ~ ∈ C β ( R × T d ) \widetilde{f} \in\mathcal{C}^\beta(\mathbb{R} \times\mathbb{T}^d) f ∈ C β ( R × T d ) f f f R × T d \mathbb{R} \times\mathbb{T}^d R × T d D T D_T D T
∥ f ∥ C β ( D T ) : = inf f ~ ∈ C β ( R × T d ) f ~ ∣ D T = f ∥ f ~ ∥ C β ( R × T d ) . \|f\|_{\mathcal{C}^{\beta}(D_T)} := \inf_{\substack{\widetilde{f}\in\mathcal{C}^{\beta}(\mathbb{R}\times\mathbb{T}^d)\\ \widetilde{f}|_{D_T}=f}} \|\widetilde{f}\|_{\mathcal{C}^{\beta}(\mathbb{R}\times\mathbb{T}^d)}. ∥ f ∥ C β ( D T ) := f ∈ C β ( R × T d ) f ∣ D T = f inf ∥ f ∥ C β ( R × T d ) . For distributions, restriction is understood on the interior of the cylinder; whenever the regularity is positive, the continuous representative supplies the boundary values.
Here, a periodic distribution f f f Λ T \Lambda_T Λ T R × T d \mathbb{R} \times\mathbb{T}^d R × T d
∥ f ∣ D T ∥ C β ( D T ) ≤ C β , T ∥ f ∥ C β ( Λ T ) . \|f|_{D_T}\|_{\mathcal{C}^{\beta}(D_T)} \le C_{\beta,T}\|f\|_{\mathcal{C}^{\beta}(\Lambda_T)}. ∥ f ∣ D T ∥ C β ( D T ) ≤ C β , T ∥ f ∥ C β ( Λ T ) . For β < 0 \beta< 0 β < 0 C β \mathcal{C}^{\beta} C β 0 < β < 1 0 < \beta< 1 0 < β < 1 B ∞ , ∞ β , s B_{\infty,\infty}^{\beta,s} B ∞ , ∞ β , s C s β \mathcal{C}_s^\beta C s β [7 , 42 ] . For β ≥ 1 \beta\ge1 β ≥ 1 d s d_s d s
C β ( D T ) ↪ C s γ ( D T ) , 0 < γ < min { 1 , β } . \mathcal{C}^{\beta}(D_T) \hookrightarrow\mathcal{C}_s^\gamma(D_T),\qquad0 < \gamma< \min\{1,\beta\}. C β ( D T ) ↪ C s γ ( D T ) , 0 < γ < min { 1 , β } . Moreover, L ∞ ( D T ) ↪ C − κ ( D T ) L^\infty(D_T) \hookrightarrow\mathcal{C}^{-\kappa}(D_T) L ∞ ( D T ) ↪ C − κ ( D T ) κ > 0 \kappa> 0 κ > 0 f f f L 1 L^1 L 1
∥ f ~ ∥ L ∞ ≤ C ∥ f ∥ L ∞ ( D T ) , \|\widetilde{f}\|_{L^\infty} \le C\|f\|_{L^\infty(D_T)}, ∥ f ∥ L ∞ ≤ C ∥ f ∥ L ∞ ( D T ) , and hence
∥ f ~ ∥ C − κ ( R × T d ) ≤ C ∥ f ∥ L ∞ ( D T ) . \|\widetilde{f}\|_{\mathcal{C}^{-\kappa}(\mathbb{R}\times\mathbb{T}^d)} \le C\|f\|_{L^\infty(D_T)}. ∥ f ∥ C − κ ( R × T d ) ≤ C ∥ f ∥ L ∞ ( D T ) . These properties and the equivalence of the resulting spaces under changes of smooth dyadic decompositions, are standard; see [4 , 8 ] .
For spatial regularity, we use the same construction with χ x ( k ) = ς ( ∣ k ∣ ) \chi^x(k)=\varsigma(|k|) χ x ( k ) = ς ( ∣ k ∣ ) Δ j x \Delta_j^x Δ j x
C x s : = B ∞ , ∞ s ( T d ) , ∥ f ∥ C x s : = sup j ≥ − 1 2 j s ∥ Δ j x f ∥ ∞ , C g , x s : = C ∞ ( T d ) ‾ C x s . \mathcal{C}_x^s := B_{\infty,\infty}^s(\mathbb{T}^d),\qquad\|f\|_{\mathcal{C}_x^s} := \sup_{j\ge-1} 2^{js}\|\Delta_j^x f\|_\infty,\qquad\mathcal{C}_{g,x}^s := \overline{C^\infty(\mathbb{T}^d)}^{\mathcal{C}_x^s}. C x s := B ∞ , ∞ s ( T d ) , ∥ f ∥ C x s := j ≥ − 1 sup 2 j s ∥ Δ j x f ∥ ∞ , C g , x s := C ∞ ( T d ) C x s . The spatial cutoff Π J 0 x \Pi_{J_0}^x Π J 0 x m J 0 x ( k ) = ς ( 2 − ( J 0 + 1 ) ∣ k ∣ ) m_{J_0}^x(k)=\varsigma(2^{-(J_0+1)}|k|) m J 0 x ( k ) = ς ( 2 − ( J 0 + 1 ) ∣ k ∣ ) § 5 :
P u ^ ( k ) : = ( I − k ⊗ k ∣ k ∣ 2 ) u ^ ( k ) ( k ≠ 0 ) , P u ^ ( 0 ) : = 0. \widehat{\mathcal{P}u}(k):=\left(I-\frac{k\otimes k}{|k|^2}\right)\widehat{u}(k)\quad(k\ne0),\qquad\widehat{\mathcal{P}u}(0):=0. P u ( k ) := ( I − ∣ k ∣ 2 k ⊗ k ) u ( k ) ( k = 0 ) , P u ( 0 ) := 0. It projects onto divergence-free, mean-zero fields.
Throughout, C , c > 0 C,c>0 C , c > 0 A ≲ B A\lesssim B A ≲ B A ≤ C B A\le CB A ≤ C B A ≍ B A\asymp B A ≍ B
Gaussian forcing and tail estimates
All random objects are defined on a complete probability space ( Ω , F , P ) (\Omega,\mathcal{F},\mathbb{P}) ( Ω , F , P ) E \mathbb{E} E
Assumption 2.2 (Gaussian forcing). There is a β 0 ∈ ( − 2 , 0 ) \beta_0\in(-2,0) β 0 ∈ ( − 2 , 0 ) η \eta η X 0 : = C g β 0 ( Λ T ) X_0:=C_g^{\beta_0}(\Lambda_T) X 0 := C g β 0 ( Λ T )
We use a weaker norm to measure perturbations of the forcing. Fix
− 2 < β 1 < β 0 < 0 , δ : = β 0 − β 1 > 0 , X 1 : = C β 1 ( D T ) , Y : = C ( D T ) . (2) -2<\beta_1<\beta_0<0,\qquad\delta:=\beta_0-\beta_1>0,\qquad X_1:=C^{\beta_1}(D_T),\qquad Y:=C(D_T). \tag*{(2)} − 2 < β 1 < β 0 < 0 , δ := β 0 − β 1 > 0 , X 1 := C β 1 ( D T ) , Y := C ( D T ) . (2) Restriction followed by the change of regularity gives a continuous map X 0 → X 1 X_0\to X_1 X 0 → X 1 Y Y Y ∥ u ∥ Y = sup ( t , x ) ∈ D T ∣ u ( t , x ) ∣ \|u\|_Y=\sup_{(t,x)\in D_T}|u(t,x)| ∥ u ∥ Y = sup ( t , x ) ∈ D T ∣ u ( t , x ) ∣ δ \delta δ β 1 > − 2 \beta_1>-2 β 1 > − 2
Lemma 2.3 (Gaussian tails). Under Assumption 2.2, there are c , C > 0 c,C>0 c , C > 0
E exp ( c ∥ η ∥ X 0 2 ) < ∞ , P ( ∥ η ∥ X 0 > R ) ≤ C e − c R 2 , R ≥ 0. \mathbb{E}\exp(c\|\eta\|_{X_0}^2)<\infty,\qquad\mathbb{P}(\|\eta\|_{X_0}>R)\le Ce^{-cR^2},\qquad R\ge0. E exp ( c ∥ η ∥ X 0 2 ) < ∞ , P ( ∥ η ∥ X 0 > R ) ≤ C e − c R 2 , R ≥ 0. In particular, E ∥ η ∥ X 0 p < ∞ \mathbb{E}\|\eta\|_{X_0}^p<\infty E ∥ η ∥ X 0 p < ∞ 0 < p < ∞ 0<p<\infty 0 < p < ∞ q ≥ 0 q\ge0 q ≥ 0 c q , C q > 0 c_q,C_q>0 c q , C q > 0 R R R
E [ ( 1 + ∥ η ∥ X 0 ) q 1 { ∥ η ∥ X 0 > R } ] ≤ C q e − c q R 2 , R ≥ 0. (3) \mathbb{E}\left[(1+\|\eta\|_{X_0})^q\mathbf{1}_{\{\|\eta\|_{X_0}>R\}}\right]\le C_qe^{-c_qR^2},\qquad R\ge0. \tag*{(3)} E [ ( 1 + ∥ η ∥ X 0 ) q 1 { ∥ η ∥ X 0 > R } ] ≤ C q e − c q R 2 , R ≥ 0. (3) Proof. Write Z = ∥ η ∥ X 0 Z=\|\eta\|_{X_0} Z = ∥ η ∥ X 0 a > 0 a>0 a > 0 E e a Z 2 < ∞ \mathbb{E}e^{aZ^2}<\infty E e a Z 2 < ∞ [6 Chapter 2] . By Markov’s inequality applied to e a Z 2 e^{aZ^2} e a Z 2 R > 0 R>0 R > 0
P ( Z > R ) = P ( e a Z 2 > e a R 2 ) ≤ e − a R 2 E e a Z 2 . \mathbb{P}(Z>R)=\mathbb{P}(e^{aZ^2}>e^{aR^2})\le e^{-aR^2}\mathbb{E}e^{aZ^2}. P ( Z > R ) = P ( e a Z 2 > e a R 2 ) ≤ e − a R 2 E e a Z 2 . Moreover, since ∣ z ∣ p ≤ C p , a e a z 2 |z|^p\le C_{p,a}e^{az^2} ∣ z ∣ p ≤ C p , a e a z 2 p > 0 p>0 p > 0 C p , a < ∞ C_{p,a}<\infty C p , a < ∞ E e a Z 2 \mathbb{E}e^{aZ^2} E e a Z 2
E ∣ Z ∣ p ≤ C p , a E e a Z 2 < ∞ . \mathbb{E}|Z|^p\le C_{p,a}\mathbb{E}e^{aZ^2}<\infty. E ∣ Z ∣ p ≤ C p , a E e a Z 2 < ∞. For the weighted tail, fix q ≥ 0 q\ge0 q ≥ 0 C q C_q C q ( 1 + z ) q ≤ C q e a z 2 / 2 (1+z)^q\le C_qe^{az^2/2} ( 1 + z ) q ≤ C q e a z 2 /2 z ≥ 0 z\ge0 z ≥ 0 { Z > R } \{Z>R\} { Z > R }
( 1 + Z ) q ≤ C q e a Z 2 e − a R 2 / 2 . (1+Z)^q\le C_qe^{aZ^2}e^{-aR^2/2}. ( 1 + Z ) q ≤ C q e a Z 2 e − a R 2 /2 . Taking expectations gives (3), with c q = a / 2 c_q=a/2 c q = a /2 C q C_q C q R R R □ \square □
The heat equation and the Schauder estimate
Write S ( t ) = e t Δ S(t)=e^{t\Delta} S ( t ) = e t Δ
∥ S ( t ) f ∥ ∞ ≤ ∥ f ∥ ∞ , t ≥ 0. \lVert S(t)f\rVert_{\infty}\leq\lVert f\rVert_{\infty},\qquad t\geq0. ∥ S ( t ) f ∥ ∞ ≤ ∥ f ∥ ∞ , t ≥ 0. For smooth forcing ϕ \phi ϕ
K ϕ ( t ) : = ∫ 0 t S ( t − r ) ϕ ( r ) d r . K\phi(t):=\int_{0}^{t}S(t-r)\phi(r)\,\mathrm{d}r. K ϕ ( t ) := ∫ 0 t S ( t − r ) ϕ ( r ) d r . For distributional forcing, the integral is interpreted through the zero-initial-data heat equation, as justified in the next lemma. With viscosity ν > 0 \nu>0 ν > 0 K ν K_{\nu} K ν S ( t − r ) S(t-r) S ( t − r ) e ν ( t − r ) Δ e^{\nu(t-r)\Delta} e ν ( t − r ) Δ
Lemma 2.4 (Schauder estimate). Let − 2 < β < 0 -2<\beta<0 − 2 < β < 0 ( ∂ t − Δ ) v = ϕ (\partial_t-\Delta)v=\phi ( ∂ t − Δ ) v = ϕ v ( 0 ) = 0 v(0)=0 v ( 0 ) = 0 K : C β ( D T ) → C β + 2 ( D T ) K:C^{\beta}(D_T)\to C^{\beta+2}(D_T) K : C β ( D T ) → C β + 2 ( D T ) v = K ϕ v=K\phi v = K ϕ
∥ K ϕ ∥ C β + 2 ( D T ) ≤ C β ∥ ϕ ∥ C β ( D T ) . \lVert K\phi\rVert_{C^{\beta+2}(D_T)}\leq C_{\beta}\lVert\phi\rVert_{C^{\beta}(D_T)}. ∥ K ϕ ∥ C β + 2 ( D T ) ≤ C β ∥ ϕ ∥ C β ( D T ) . In particular, K ϕ K\phi K ϕ 0 < γ < min { 1 , β + 2 } 0<\gamma<\min\{1,\beta+2\} 0 < γ < min { 1 , β + 2 }
∥ K ϕ ∥ C γ ( D T ) ≤ C β , γ ∥ ϕ ∥ C β ( D T ) . (4) \lVert K\phi\rVert_{C^{\gamma}(D_T)}\leq C_{\beta,\gamma}\lVert\phi\rVert_{C^{\beta}(D_T)}. \tag*{(4)} ∥ K ϕ ∥ C γ ( D T ) ≤ C β , γ ∥ ϕ ∥ C β ( D T ) . (4) The same assertions hold for K ν K_{\nu} K ν ν > 0 \nu>0 ν > 0
Proof. Set α = β + 2 ∈ ( 0 , 2 ) \alpha=\beta+2\in(0,2) α = β + 2 ∈ ( 0 , 2 ) ϕ ~ ∈ C β ( R × T d ) \widetilde{\phi}\in C^{\beta}(\mathbb{R}\times\mathbb{T}^{d}) ϕ ∈ C β ( R × T d ) ϕ \phi ϕ
∥ ϕ ~ ∥ C β ( R × T d ) ≤ 2 ∥ ϕ ∥ C β ( D T ) . \lVert\widetilde{\phi}\rVert_{C^{\beta}(\mathbb{R}\times\mathbb{T}^{d})}\leq2\lVert\phi\rVert_{C^{\beta}(D_T)}. ∥ ϕ ∥ C β ( R × T d ) ≤ 2 ∥ ϕ ∥ C β ( D T ) . Let ϑ ∈ C c ∞ ( R ) \vartheta\in C^{\infty}_{c}(\mathbb{R}) ϑ ∈ C c ∞ ( R ) ϑ ≡ 1 \vartheta\equiv1 ϑ ≡ 1 [ 0 , T ] [0,T] [ 0 , T ]
g ( t , x ) : = ϑ ( t ) e − t ϕ ~ ( t , x ) , ( t , x ) ∈ R × T d . g(t,x):=\vartheta(t)e^{-t}\widetilde{\phi}(t,x),\qquad(t,x)\in\mathbb{R}\times\mathbb{T}^{d}. g ( t , x ) := ϑ ( t ) e − t ϕ ( t , x ) , ( t , x ) ∈ R × T d . Multiplication by this fixed smooth function is bounded on C β C^{\beta} C β
∥ g ∥ C β ( R × T d ) ≤ C T ∥ ϕ ∥ C β ( D T ) . \lVert g\rVert_{C^{\beta}(\mathbb{R}\times\mathbb{T}^{d})}\leq C_T\lVert\phi\rVert_{C^{\beta}(D_T)}. ∥ g ∥ C β ( R × T d ) ≤ C T ∥ ϕ ∥ C β ( D T ) . On the full cylinder R × T d \mathbb{R}\times\mathbb{T}^{d} R × T d w = ( ∂ t − Δ + 1 ) − 1 g w=(\partial_t-\Delta+1)^{-1}g w = ( ∂ t − Δ + 1 ) − 1 g ( 1 + i ω + ∣ k ∣ 2 ) − 1 (1+i\omega+\lvert k\rvert^2)^{-1} ( 1 + iω + ∣ k ∣ 2 ) − 1 j ≥ 0 j\geq0 j ≥ 0 Δ j s \Delta_j^s Δ j s ∣ ω ∣ ∼ 2 2 j \lvert\omega\rvert\sim2^{2j} ∣ ω ∣ ∼ 2 2 j ∣ k ∣ ∼ 2 j \lvert k\rvert\sim2^j ∣ k ∣ ∼ 2 j
∣ 1 + i ω + ∣ k ∣ 2 ∣ ≳ 2 2 j , \lvert1+i\omega+\lvert k\rvert^2\rvert\gtrsim2^{2j}, ∣ 1 + iω + ∣ k ∣ 2 ∣ ≳ 2 2 j , so the localized multiplier has convolution kernel with L 1 L^1 L 1 C 2 − 2 j C2^{-2j} C 2 − 2 j
∥ Δ j s w ∥ ∞ ≤ C 2 − 2 j ∑ ℓ ≥ − 1 ∣ ℓ − j ∣ ≤ C ∥ Δ ℓ s g ∥ ∞ . \lVert\Delta_j^s w\rVert_{\infty}\leq C2^{-2j}\sum_{\substack{\ell\geq-1\\ \lvert\ell-j\rvert\leq C}}\lVert\Delta_\ell^s g\rVert_{\infty}. ∥ Δ j s w ∥ ∞ ≤ C 2 − 2 j ℓ ≥ − 1 ∣ ℓ − j ∣ ≤ C ∑ ∥ Δ ℓ s g ∥ ∞ . For the low-frequency block j = − 1 j=-1 j = − 1 ∣ ω ∣ ≲ 1 \lvert\omega\rvert\lesssim1 ∣ ω ∣ ≲ 1 ∣ k ∣ ≲ 1 \lvert k\rvert\lesssim1 ∣ k ∣ ≲ 1 1 1 1
∣ 1 + i ω + ∣ k ∣ 2 ∣ ≳ 1 , \lvert1+i\omega+\lvert k\rvert^2\rvert\gtrsim1, ∣ 1 + iω + ∣ k ∣ 2 ∣ ≳ 1 , so the multiplier ( 1 + i ω + ∣ k ∣ 2 ) − 1 (1+i\omega+\lvert k\rvert^2)^{-1} ( 1 + iω + ∣ k ∣ 2 ) − 1 L 1 L^1 L 1
∥ Δ − 1 s w ∥ ∞ ≤ C ∑ ℓ ≥ − 1 ∣ ℓ + 1 ∣ ≤ C ∥ Δ ℓ s g ∥ ∞ . \lVert\Delta_{-1}^s w\rVert_{\infty}\leq C\sum_{\substack{\ell\geq-1\\ \lvert\ell+1\rvert\leq C}}\lVert\Delta_\ell^s g\rVert_{\infty}. ∥ Δ − 1 s w ∥ ∞ ≤ C ℓ ≥ − 1 ∣ ℓ + 1 ∣ ≤ C ∑ ∥ Δ ℓ s g ∥ ∞ . By the dyadic characterization of C α C^{\alpha} C α α − 2 = β \alpha-2=\beta α − 2 = β j ≥ 0 j\geq0 j ≥ 0
2 j α ∥ Δ j s w ∥ ∞ ≲ ∑ ∣ ℓ − j ∣ ≤ N 2 ( j − ℓ ) β ( 2 ℓ β ∥ Δ ℓ s g ∥ ∞ ) ≲ ∥ g ∥ C β , (5) 2^{j\alpha}\lVert\Delta_j^s w\rVert_{\infty}\lesssim\sum_{\lvert\ell-j\rvert\leq N}2^{(j-\ell)\beta}\left(2^{\ell\beta}\lVert\Delta_\ell^s g\rVert_{\infty}\right)\lesssim\lVert g\rVert_{C^{\beta}}, \tag*{(5)} 2 j α ∥ Δ j s w ∥ ∞ ≲ ∣ ℓ − j ∣ ≤ N ∑ 2 ( j − ℓ ) β ( 2 ℓ β ∥ Δ ℓ s g ∥ ∞ ) ≲ ∥ g ∥ C β , (5) since the sum contains only finitely many terms and ∣ j − ℓ ∣ ≤ C |j-\ell| \le C ∣ j − ℓ ∣ ≤ C j ≥ − 1 j \ge-1 j ≥ − 1
∥ w ∥ C α ( R × T d ) ≤ C ∥ g ∥ C β ( R × T d ) . \|w\|_{C^\alpha(\mathbb{R}\times\mathbb{T}^d)} \le C\|g\|_{C^\beta(\mathbb{R}\times\mathbb{T}^d)}. ∥ w ∥ C α ( R × T d ) ≤ C ∥ g ∥ C β ( R × T d ) . Since α > 0 \alpha> 0 α > 0 w w w t = 0 t=0 t = 0
∥ w ( 0 ) ∥ C x α ≤ C ∥ w ∥ C α ( R × T d ) . \|w(0)\|_{C^\alpha_x} \le C\|w\|_{C^\alpha(\mathbb{R}\times\mathbb{T}^d)}. ∥ w ( 0 ) ∥ C x α ≤ C ∥ w ∥ C α ( R × T d ) . Recall the standard estimate for heat extension (see [4 , 8 , 15 ] )
∥ S ( ⋅ ) f ∥ C α ( D T ) ≤ C ∥ f ∥ C x α , 0 < α < 2. \|S(\cdot)f\|_{C^\alpha(D_T)} \le C\|f\|_{C^\alpha_x}, \qquad0 < \alpha< 2. ∥ S ( ⋅ ) f ∥ C α ( D T ) ≤ C ∥ f ∥ C x α , 0 < α < 2. Note that for α = 1 \alpha=1 α = 1
Now define
v ( t ) : = e t w ( t ) − S ( t ) w ( 0 ) , 0 ≤ t ≤ T . v(t) := e^t w(t) - S(t)w(0), \qquad0 \le t \le T. v ( t ) := e t w ( t ) − S ( t ) w ( 0 ) , 0 ≤ t ≤ T . On ( 0 , T ) × T d (0,T)\times\mathbb{T}^d ( 0 , T ) × T d ( ∂ t − Δ ) v = e t g = φ (\partial_t-\Delta)v=e^t g=\varphi ( ∂ t − Δ ) v = e t g = φ v ( 0 ) = 0 v(0)=0 v ( 0 ) = 0
∥ v ∥ C α ( D T ) ≤ C ∥ φ ∥ C β ( D T ) . \|v\|_{C^\alpha(D_T)} \le C\|\varphi\|_{C^\beta(D_T)}. ∥ v ∥ C α ( D T ) ≤ C ∥ φ ∥ C β ( D T ) . Replacing ∣ k ∣ 2 |k|^2 ∣ k ∣ 2 ν ∣ k ∣ 2 \nu|k|^2 ν ∣ k ∣ 2
The semilinear solution map
Let F : R → R F:\mathbb{R}\to\mathbb{R} F : R → R L F L_F L F φ ∈ X 1 = C β 1 ( D T ) \varphi\in X_1=C^{\beta_1}(D_T) φ ∈ X 1 = C β 1 ( D T ) u ∈ Y = C ( D T ) u\in Y=C(D_T) u ∈ Y = C ( D T )
u ( t ) = S ( t ) u 0 + K φ ( t ) + ∫ 0 t S ( t − r ) F ( u ( r ) ) d r . (6) u(t)=S(t)u_0+K\varphi(t)+\int_0^t S(t-r)F(u(r))\,dr. \tag*{(6)} u ( t ) = S ( t ) u 0 + K φ ( t ) + ∫ 0 t S ( t − r ) F ( u ( r )) d r . (6) Theorem 2.5 (Well-posedness and pathwise stability). For every ( u 0 , φ ) ∈ C ( T d ) × X 1 (u_0,\varphi)\in C(\mathbb{T}^d)\times X_1 ( u 0 , φ ) ∈ C ( T d ) × X 1 there is a unique mild solution u = S ( u 0 , φ ) ∈ Y u=S(u_0,\varphi)\in Y u = S ( u 0 , φ ) ∈ Y For two pairs of data,
∥ S ( u 0 1 , φ 1 ) − S ( u 0 2 , φ 2 ) ∥ Y ≤ e L F T ( ∥ u 0 1 − u 0 2 ∥ ∞ + C β 1 ∥ φ 1 − φ 2 ∥ X 1 ) . (7) \|S(u_0^1,\varphi^1)-S(u_0^2,\varphi^2)\|_Y
\le e^{L_FT}\left(\|u_0^1-u_0^2\|_\infty+C_{\beta_1}\|\varphi^1-\varphi^2\|_{X_1}\right). \tag*{(7)} ∥ S ( u 0 1 , φ 1 ) − S ( u 0 2 , φ 2 ) ∥ Y ≤ e L F T ( ∥ u 0 1 − u 0 2 ∥ ∞ + C β 1 ∥ φ 1 − φ 2 ∥ X 1 ) . (7) Moreover,
∥ S ( u 0 , φ ) ∥ Y ≤ e L F T ( ∥ u 0 ∥ ∞ + C β 1 ∥ φ ∥ X 1 + T ∣ F ( 0 ) ∣ ) . (8) \|S(u_0,\varphi)\|_Y\le e^{L_FT}\left(\|u_0\|_\infty+C_{\beta_1}\|\varphi\|_{X_1}+T|F(0)|\right). \tag*{(8)} ∥ S ( u 0 , φ ) ∥ Y ≤ e L F T ( ∥ u 0 ∥ ∞ + C β 1 ∥ φ ∥ X 1 + T ∣ F ( 0 ) ∣ ) . (8) Proof. By lemma 2.4, we have K φ ∈ Y K\varphi\in Y K φ ∈ Y S : Y → Y S:Y\to Y S : Y → Y Y Y Y
∥ u ∥ a : = sup 0 ≤ t ≤ T e − a t ∥ u ( t ) ∥ ∞ , a : = 2 L F + 1. \|u\|_a:=\sup_{0\le t\le T}e^{-at}\|u(t)\|_\infty,\qquad a:=2L_F+1. ∥ u ∥ a := 0 ≤ t ≤ T sup e − a t ∥ u ( t ) ∥ ∞ , a := 2 L F + 1. Direct computation gives
e − a t ∥ S u ( t ) − S v ( t ) ∥ ∞ ≤ L F ∫ 0 t e − a ( t − r ) d r ∥ u − v ∥ a ≤ L F a ∥ u − v ∥ a . e^{-at}\|S u(t)-S v(t)\|_\infty\le L_F\int_0^t e^{-a(t-r)}\,dr\,\|u-v\|_a\le\frac{L_F}{a}\|u-v\|_a. e − a t ∥ S u ( t ) − S v ( t ) ∥ ∞ ≤ L F ∫ 0 t e − a ( t − r ) d r ∥ u − v ∥ a ≤ a L F ∥ u − v ∥ a . Since L F / a < 1 / 2 L_F/a<1/2 L F / a < 1/2 [ 0 , T ] [0,T] [ 0 , T ]
Now for u i = S ( u 0 i , φ i ) u^i=S(u_0^i,\varphi^i) u i = S ( u 0 i , φ i )
∥ u 1 ( t ) − u 2 ( t ) ∥ ∞ ≤ A + L F ∫ 0 t ∥ u 1 ( r ) − u 2 ( r ) ∥ ∞ d r , \|u^1(t)-u^2(t)\|_\infty\le A+L_F\int_0^t\|u^1(r)-u^2(r)\|_\infty\,dr, ∥ u 1 ( t ) − u 2 ( t ) ∥ ∞ ≤ A + L F ∫ 0 t ∥ u 1 ( r ) − u 2 ( r ) ∥ ∞ d r , where A = ∥ u 0 1 − u 0 2 ∥ ∞ + C β 1 ∥ φ 1 − φ 2 ∥ X 1 A=\|u_0^1-u_0^2\|_\infty+C_{\beta_1}\|\varphi^1-\varphi^2\|_{X_1} A = ∥ u 0 1 − u 0 2 ∥ ∞ + C β 1 ∥ φ 1 − φ 2 ∥ X 1 ∥ u 1 ( t ) − u 2 ( t ) ∥ ∞ ≤ A e L F t \|u^1(t)-u^2(t)\|_\infty\le Ae^{L_Ft} ∥ u 1 ( t ) − u 2 ( t ) ∥ ∞ ≤ A e L F t t t t ∣ F ( r ) ∣ ≤ ∣ F ( 0 ) ∣ + L F ∣ r ∣ |F(r)|\le|F(0)|+L_F|r| ∣ F ( r ) ∣ ≤ ∣ F ( 0 ) ∣ + L F ∣ r ∣
∥ u ( t ) ∥ ∞ ≤ ∥ u 0 ∥ ∞ + C β 1 ∥ φ ∥ X 1 + T ∣ F ( 0 ) ∣ + L F ∫ 0 t ∥ u ( r ) ∥ ∞ d r . \|u(t)\|_\infty\le\|u_0\|_\infty+C_{\beta_1}\|\varphi\|_{X_1}+T|F(0)|+L_F\int_0^t\|u(r)\|_\infty\,dr. ∥ u ( t ) ∥ ∞ ≤ ∥ u 0 ∥ ∞ + C β 1 ∥ φ ∥ X 1 + T ∣ F ( 0 ) ∣ + L F ∫ 0 t ∥ u ( r ) ∥ ∞ d r . Then (7) follows from the Gronwall inequality. □
Uniform bounds on bounded sets of data
Assumption 2.6 (Initial-data class). For fixed M , L 0 > 0 M, L_0 > 0 M , L 0 > 0
G 0 : = { u 0 ∈ C ( T d ) : ∥ u 0 ∥ ∞ ≤ M , ∣ u 0 ( x ) − u 0 ( y ) ∣ ≤ L 0 d T d ( x , y ) for all x , y ∈ T d } . \mathcal{G}_0 := \left\{u_0 \in C(\mathbb{T}^d) : \|u_0\|_\infty\le M,\quad|u_0(x)-u_0(y)| \le L_0d_{\mathbb{T}^d}(x,y)\ \text{for all }x,y\in\mathbb{T}^d\right\}. G 0 := { u 0 ∈ C ( T d ) : ∥ u 0 ∥ ∞ ≤ M , ∣ u 0 ( x ) − u 0 ( y ) ∣ ≤ L 0 d T d ( x , y ) for all x , y ∈ T d } . For the approximation of the output functions, we also need a modulus of continuity that is uniform over bounded forcing sets.
Lemma 2.7 (Uniform solution bounds). Fix
0 < γ < min { 1 , β 0 + 2 } . (9) 0 < \gamma< \min\{1,\beta_0+2\}. \tag*{(9)} 0 < γ < min { 1 , β 0 + 2 } . (9) There are increasing affine functions B , Λ : [ 0 , ∞ ) → [ 1 , ∞ ) B,\Lambda:[0,\infty)\to[1,\infty) B , Λ : [ 0 , ∞ ) → [ 1 , ∞ ) d d d T T T β 0 \beta_0 β 0 β 1 \beta_1 β 1 γ \gamma γ M M M L 0 L_0 L 0 L F L_F L F ∣ F ( 0 ) ∣ |F(0)| ∣ F ( 0 ) ∣ R ≥ 0 R\ge0 R ≥ 0 u 0 ∈ G 0 u_0\in\mathcal{G}_0 u 0 ∈ G 0 ζ ∈ X 0 \zeta\in X_0 ζ ∈ X 0 ∥ ζ ∥ X 0 ≤ R \|\zeta\|_{X_0}\le R ∥ ζ ∥ X 0 ≤ R
∥ S ( u 0 , ζ ) ∥ Y ≤ B ( R ) , \|S(u_0,\zeta)\|_Y \le B(R), ∥ S ( u 0 , ζ ) ∥ Y ≤ B ( R ) , ∣ S ( u 0 , ζ ) ( z ) − S ( u 0 , ζ ) ( z ′ ) ∣ ≤ Λ ( R ) d s ( z , z ′ ) γ . |S(u_0,\zeta)(z)-S(u_0,\zeta)(z')| \le\Lambda(R)d_s(z,z')^\gamma. ∣ S ( u 0 , ζ ) ( z ) − S ( u 0 , ζ ) ( z ′ ) ∣ ≤ Λ ( R ) d s ( z , z ′ ) γ . In particular, B ( R ) + Λ ( R ) ≤ C ( 1 + R ) B(R)+\Lambda(R)\le C(1+R) B ( R ) + Λ ( R ) ≤ C ( 1 + R )
Proof. Let C r e s C_{\mathrm{res}} C res X 0 X_0 X 0 X 1 X_1 X 1 (8) gives the first assertion with
B ( R ) : = 1 + e L F T ( M + C β 1 C r e s R + T ∣ F ( 0 ) ∣ ) . B(R):=1+e^{L_FT}\left(M+C\beta_1C_{\mathrm{res}}R+T|F(0)|\right). B ( R ) := 1 + e L F T ( M + C β 1 C res R + T ∣ F ( 0 ) ∣ ) . Let L 0 = Lip ( u 0 ) L_0=\operatorname{Lip}(u_0) L 0 = Lip ( u 0 ) Lip ( S ( t ) u 0 ) ≤ L 0 \operatorname{Lip}(S(t)u_0)\le L_0 Lip ( S ( t ) u 0 ) ≤ L 0
∥ S ( h ) u 0 − u 0 ∥ ∞ ≤ L 0 ∫ T d K h ( z ) d T d ( z , 0 ) d z ≤ C L 0 h 1 / 2 . \begin{aligned}
\|S(h)u_0-u_0\|_\infty\le L_0\int_{\mathbb{T}^d}K_h(z)d_{\mathbb{T}^d}(z,0)\,dz \\
&\le CL_0h^{1/2}.
\end{aligned} ∥ S ( h ) u 0 − u 0 ∥ ∞ ≤ L 0 ∫ T d K h ( z ) d T d ( z , 0 ) d z ≤ C L 0 h 1/2 . For t ≥ s ≥ 0 t\ge s\ge0 t ≥ s ≥ 0
∣ S ( t ) u 0 ( x ) − S ( s ) u 0 ( y ) ∣ ≤ ∣ S ( t ) u 0 ( x ) − S ( t ) u 0 ( y ) ∣ + ∥ S ( s ) ( S ( t − s ) u 0 − u 0 ) ∥ ∞ ≤ L 0 d T d ( x , y ) + ∥ S ( t − s ) u 0 − u 0 ∥ ∞ ≤ C L 0 ( d T d ( x , y ) + ∣ t − s ∣ 1 / 2 ) . \begin{aligned}
|S(t)u_0(x)-S(s)u_0(y)|\le|S(t)u_0(x)-S(t)u_0(y)|+\|S(s)(S(t-s)u_0-u_0)\|_\infty\\
&\le L_0d_{\mathbb{T}^d}(x,y)+\|S(t-s)u_0-u_0\|_\infty\\
&\le CL_0\left(d_{\mathbb{T}^d}(x,y)+|t-s|^{1/2}\right).
\end{aligned} ∣ S ( t ) u 0 ( x ) − S ( s ) u 0 ( y ) ∣ ≤ ∣ S ( t ) u 0 ( x ) − S ( t ) u 0 ( y ) ∣ + ∥ S ( s ) ( S ( t − s ) u 0 − u 0 ) ∥ ∞ ≤ L 0 d T d ( x , y ) + ∥ S ( t − s ) u 0 − u 0 ∥ ∞ ≤ C L 0 ( d T d ( x , y ) + ∣ t − s ∣ 1/2 ) . Writing D = diam s ( D T ) < ∞ D=\operatorname{diam}_s(D_T)<\infty D = diam s ( D T ) < ∞ 0 < γ < 1 0<\gamma<1 0 < γ < 1
[ S ( ⋅ ) u 0 ] C s γ ( D T ) ≤ D 1 − γ [ S ( ⋅ ) u 0 ] C s 1 ( D T ) ≤ C L 0 , ∥ S ( ⋅ ) u 0 ∥ ∞ ≤ M . [S(\cdot)u_0]_{C_s^\gamma(D_T)}\le D^{1-\gamma}[S(\cdot)u_0]_{C_s^1(D_T)}\le CL_0,\qquad\|S(\cdot)u_0\|_\infty\le M. [ S ( ⋅ ) u 0 ] C s γ ( D T ) ≤ D 1 − γ [ S ( ⋅ ) u 0 ] C s 1 ( D T ) ≤ C L 0 , ∥ S ( ⋅ ) u 0 ∥ ∞ ≤ M . Hence
∥ S ( ⋅ ) u 0 ∥ C s γ ( D T ) ≤ C ( M + L 0 ) , \|S(\cdot)u_0\|_{C_s^\gamma(D_T)}\le C(M+L_0), ∥ S ( ⋅ ) u 0 ∥ C s γ ( D T ) ≤ C ( M + L 0 ) , uniformly up to t = 0 t=0 t = 0
By Lemma 2.4 with β = β 0 \beta=\beta_0 β = β 0 γ < β 0 + 2 \gamma<\beta_0+2 γ < β 0 + 2
∥ K ζ ∥ C s γ ( D T ) ≤ C ∥ ζ ∥ C β 0 ( D T ) ≤ C R . \|K\zeta\|_{C_s^\gamma(D_T)}\le C\|\zeta\|_{C^{\beta_0}(D_T)}\le CR. ∥ K ζ ∥ C s γ ( D T ) ≤ C ∥ ζ ∥ C β 0 ( D T ) ≤ C R . Moreover,
∥ F ( u ) ∥ ∞ ≤ ∣ F ( 0 ) ∣ + L F ∥ u ∥ ∞ ≤ ∣ F ( 0 ) ∣ + L F B ( R ) . \|F(u)\|_\infty\le|F(0)|+L_F\|u\|_\infty\le|F(0)|+L_FB(R). ∥ F ( u ) ∥ ∞ ≤ ∣ F ( 0 ) ∣ + L F ∥ u ∥ ∞ ≤ ∣ F ( 0 ) ∣ + L F B ( R ) . Since
L ∞ ↪ C − 1 / 2 , γ < 1 < − 1 2 + 2 = 3 2 , L^\infty\hookrightarrow C^{-1/2},\qquad\gamma<1<-\frac{1}{2}+2=\frac{3}{2}, L ∞ ↪ C − 1/2 , γ < 1 < − 2 1 + 2 = 2 3 , another application of Lemma 2.4 gives
∥ K ( F ( u ) ) ∥ C s γ ( D T ) ≤ C ∥ F ( u ) ∥ C − 1 / 2 ( D T ) ≤ C ∥ F ( u ) ∥ ∞ ≤ C ( ∣ F ( 0 ) ∣ + L F B ( R ) ) . \begin{aligned}
\|K(F(u))\|_{C_s^\gamma(D_T)}\le C\|F(u)\|_{C^{-1/2}(D_T)} \\
&\le C\|F(u)\|_\infty\\
&\le C\left(|F(0)|+L_FB(R)\right).
\end{aligned} ∥ K ( F ( u )) ∥ C s γ ( D T ) ≤ C ∥ F ( u ) ∥ C − 1/2 ( D T ) ≤ C ∥ F ( u ) ∥ ∞ ≤ C ( ∣ F ( 0 ) ∣ + L F B ( R ) ) . Combining these estimates proves the second assertion with
Λ ( R ) : = 1 + C ( M + L 0 + R + ∣ F ( 0 ) ∣ + L F B ( R ) ) . \Lambda(R) := 1 + C\bigl(M + L_0 + R + \lvert F(0)\rvert+ L_F B(R)\bigr). Λ ( R ) := 1 + C ( M + L 0 + R + ∣ F ( 0 )∣ + L F B ( R ) ) . Both B ( R ) B(R) B ( R ) Λ ( R ) \Lambda(R) Λ ( R ) R R R □ \square □
Finite-dimensional encoding and operator architecture
A real finite-rank measurement map
Let Π J = ∑ j ≤ J Δ j s \Pi_J = \sum_{j\le J} \Delta^s_j Π J = ∑ j ≤ J Δ j s m J m_J m J
0 ≤ m J ( λ ) ≤ m J + 1 ( λ ) ≤ 1 , m J ( λ ) ⟶ 1 for every fixed λ . (10) 0 \le m_J(\lambda) \le m_{J+1}(\lambda) \le1,\qquad m_J(\lambda) \longrightarrow1\quad\text{for every fixed }\lambda. \tag*{(10)} 0 ≤ m J ( λ ) ≤ m J + 1 ( λ ) ≤ 1 , m J ( λ ) ⟶ 1 for every fixed λ . (10) The operator Π J \Pi_J Π J ( e λ ) λ (e_\lambda)_\lambda ( e λ ) λ L 2 ( Λ T ) L^2(\Lambda_T) L 2 ( Λ T ) { θ , − θ } \{\theta,-\theta\} { θ , − θ } m J ( θ ) = m J ( − θ ) m_J(\theta) = m_J(-\theta) m J ( θ ) = m J ( − θ ) Π J \Pi_J Π J m J ( θ ) m_J(\theta) m J ( θ ) e λ e_\lambda e λ Π J \Pi_J Π J
Let Λ J \Lambda_J Λ J Π J \Pi_J Π J
Λ J : = { λ : Π J e λ ≠ 0 } . \Lambda_J := \{\lambda: \Pi_J e_\lambda\ne0\}. Λ J := { λ : Π J e λ = 0 } . Define
c J ( ζ ) : = ( ⟨ ζ , e λ ⟩ ) λ ∈ Λ J ∈ R N J , Rec J ( c ) : = ∑ λ ∈ Λ J m J ( λ ) c λ e λ . (11) c_J(\zeta) := \bigl(\langle\zeta,e_\lambda\rangle\bigr)_{\lambda\in\Lambda_J} \in\mathbb{R}^{N_J},\qquad\operatorname{Rec}_J(c) := \sum_{\lambda\in\Lambda_J} m_J(\lambda)c_\lambda e_\lambda. \tag*{(11)} c J ( ζ ) := ( ⟨ ζ , e λ ⟩ ) λ ∈ Λ J ∈ R N J , Rec J ( c ) := λ ∈ Λ J ∑ m J ( λ ) c λ e λ . (11) Then Rec J ( c J ( ζ ) ) = Π J ζ \operatorname{Rec}_J(c_J(\zeta)) = \Pi_J\zeta Rec J ( c J ( ζ )) = Π J ζ
N J : = ∣ Λ J ∣ ≍ 2 J ( d + 2 ) . (12) N_J := \lvert\Lambda_J\rvert\asymp2^{J(d+2)}. \tag*{(12)} N J := ∣ Λ J ∣ ≍ 2 J ( d + 2 ) . (12) The count and the uniform boundedness of the smooth cutoffs will be proved in Lemma 3.1. For an abstract forcing satisfying Assumption 2.2, c J ( η ) c_J(\eta) c J ( η ) Σ λ μ = E [ ⟨ η , e λ ⟩ ⟨ η , e μ ⟩ ] \Sigma_{\lambda\mu} = \mathbb{E}[\langle\eta,e_\lambda\rangle\langle\eta,e_\mu\rangle] Σ λ μ = E [⟨ η , e λ ⟩ ⟨ η , e μ ⟩]
Lemma 3.1. For J ≥ 0 J \ge0 J ≥ 0
rank ( Π J ) ≍ 2 J ( d + 2 ) . (13) \operatorname{rank}(\Pi_J) \asymp2^{J(d+2)}. \tag*{(13)} rank ( Π J ) ≍ 2 J ( d + 2 ) . (13) Moreover, for every β ∈ R \beta\in\mathbb{R} β ∈ R C β > 0 C_\beta> 0 C β > 0 J J J
∥ Π J f ∥ C β ≤ C β ∥ f ∥ C β . (14) \lVert\Pi_J f\rVert_{C^\beta} \le C_\beta\lVert f\rVert_{C^\beta}. \tag*{(14)} ∥ Π J f ∥ C β ≤ C β ∥ f ∥ C β . (14) Proof. For a > 0 a > 0 a > 0
Q J ( a ) : = { ( n , k ) ∈ Z × Z d : ∣ n ∣ ≤ a 2 2 J , ∣ k ∣ ≤ a 2 J } . Q_J(a) := \{(n,k) \in\mathbb{Z}\times\mathbb{Z}^d : \lvert n\rvert\le a2^{2J},\ \lvert k\rvert\le a2^J\}. Q J ( a ) := {( n , k ) ∈ Z × Z d : ∣ n ∣ ≤ a 2 2 J , ∣ k ∣ ≤ a 2 J } . The parabolic cutoff construction gives fixed constants 0 < c < C 0 < c < C 0 < c < C
Q J ( c ) ⊆ { ( n , k ) : m J ( n , k ) ≠ 0 } ⊆ Q J ( C ) . Q_J(c) \subseteq\{(n,k) : m_J(n,k) \ne0\} \subseteq Q_J(C). Q J ( c ) ⊆ {( n , k ) : m J ( n , k ) = 0 } ⊆ Q J ( C ) . Since
∣ Q J ( a ) ∣ ≍ 2 2 J 2 d J = 2 J ( d + 2 ) , \lvert Q_J(a)\rvert\asymp2^{2J}2^{dJ} = 2^{J(d+2)}, ∣ Q J ( a )∣ ≍ 2 2 J 2 dJ = 2 J ( d + 2 ) , and the rank of Π J \Pi_J Π J m J m_J m J N N N
Δ j s Π J f = ∑ ∣ j − j ′ ∣ ≤ N Δ j s Π J Δ j ′ s f . \Delta^s_j\Pi_J f = \sum_{\lvert j-j'\rvert\le N}\Delta^s_j\Pi_J\Delta^s_{j'}f. Δ j s Π J f = ∣ j − j ′ ∣ ≤ N ∑ Δ j s Π J Δ j ′ s f . e n d s m i d = 1 {ends_mid=1} e n d s m i d = 1 The operators Δ j s Π J \Delta^s_j\Pi_J Δ j s Π J L 1 L^1 L 1 j j j J J J
2 j β ∥ Δ j s Π J f ∥ ∞ ≤ C ∑ ∣ j − j ′ ∣ ≤ N 2 ( j − j ′ ) β 2 j ′ β ∥ Δ j ′ s f ∥ ∞ ≤ C β ∥ f ∥ C β . \begin{aligned}
2^{j\beta}\|\Delta^s_j\Pi_J f\|_\infty\le C\sum_{|j-j'|\le N}2^{(j-j')\beta}2^{j'\beta}\|\Delta^s_{j'}f\|_\infty\\
&\le C_\beta\|f\|_{C^\beta}.
\end{aligned} 2 j β ∥ Δ j s Π J f ∥ ∞ ≤ C ∣ j − j ′ ∣ ≤ N ∑ 2 ( j − j ′ ) β 2 j ′ β ∥ Δ j ′ s f ∥ ∞ ≤ C β ∥ f ∥ C β . Taking the supremum over j j j □ \square □
Lemma 3.2. For every ζ ∈ C β 0 ( Λ T ) \zeta\in C^{\beta_0}(\Lambda_T) ζ ∈ C β 0 ( Λ T )
∥ ( I − Π J ) ζ ∥ C β 1 ≤ C 2 − J δ ∥ ζ ∥ C β 0 . (15) \|(I-\Pi_J)\zeta\|_{C^{\beta_1}}\le C2^{-J\delta}\|\zeta\|_{C^{\beta_0}}. \tag*{(15)} ∥ ( I − Π J ) ζ ∥ C β 1 ≤ C 2 − J δ ∥ ζ ∥ C β 0 . (15) Proof. Almost orthogonality gives Δ j s ( I − Π J ) ζ = 0 \Delta^s_j(I-\Pi_J)\zeta=0 Δ j s ( I − Π J ) ζ = 0 j < J − N j<J-N j < J − N
2 j β 1 ∥ Δ j s ( I − Π J ) ζ ∥ ∞ ≤ C 2 − j ( β 0 − β 1 ) 2 j β 0 ∥ Δ j s ζ ∥ ∞ 2^{j\beta_1}\|\Delta^s_j(I-\Pi_J)\zeta\|_\infty\le C2^{-j(\beta_0-\beta_1)}2^{j\beta_0}\|\Delta^s_j\zeta\|_\infty 2 j β 1 ∥ Δ j s ( I − Π J ) ζ ∥ ∞ ≤ C 2 − j ( β 0 − β 1 ) 2 j β 0 ∥ Δ j s ζ ∥ ∞ for j ≥ J − N j\ge J-N j ≥ J − N j j j □ \square □
Lemma 3.3. There is a constant C > 0 C>0 C > 0 J J J
∥ c J ( ζ ) ∥ ℓ ∞ ≤ C 2 − J β 0 ∥ ζ ∥ C β 0 , (16) \|c_J(\zeta)\|_{\ell^\infty}\le C2^{-J\beta_0}\|\zeta\|_{C^{\beta_0}}, \tag*{(16)} ∥ c J ( ζ ) ∥ ℓ ∞ ≤ C 2 − J β 0 ∥ ζ ∥ C β 0 , (16) ∥ Rec J ( c ) − Rec J ( c ′ ) ∥ X 1 ≤ C N J 1 / 2 ∥ c − c ′ ∥ ℓ 2 . (17) \|\operatorname{Rec}_J(c)-\operatorname{Rec}_J(c')\|_{X_1}\le CN_J^{1/2}\|c-c'\|_{\ell^2}. \tag*{(17)} ∥ Rec J ( c ) − Rec J ( c ′ ) ∥ X 1 ≤ C N J 1/2 ∥ c − c ′ ∥ ℓ 2 . (17) Proof. Let e λ e_\lambda e λ 2 j 2^j 2 j
∣ ⟨ ζ , e λ ⟩ ∣ ≤ C ∑ ∣ j ′ − j ∣ ≤ N ∥ Δ j ′ s ζ ∥ ∞ ≤ C 2 − j β 0 ∥ ζ ∥ C β 0 , |\langle\zeta,e_\lambda\rangle|\le C\sum_{|j'-j|\le N}\|\Delta^s_{j'}\zeta\|_\infty\le C2^{-j\beta_0}\|\zeta\|_{C^{\beta_0}}, ∣ ⟨ ζ , e λ ⟩ ∣ ≤ C ∣ j ′ − j ∣ ≤ N ∑ ∥ Δ j ′ s ζ ∥ ∞ ≤ C 2 − j β 0 ∥ ζ ∥ C β 0 , where N N N λ ∈ Λ J \lambda\in\Lambda_J λ ∈ Λ J j ≤ J + C 0 j\le J+C_0 j ≤ J + C 0 β 0 < 0 \beta_0<0 β 0 < 0
sup λ ∈ Λ J ∣ ⟨ ζ , e λ ⟩ ∣ ≤ C 2 − J β 0 ∥ ζ ∥ C β 0 , \sup_{\lambda\in\Lambda_J}|\langle\zeta,e_\lambda\rangle|\le C2^{-J\beta_0}\|\zeta\|_{C^{\beta_0}}, λ ∈ Λ J sup ∣ ⟨ ζ , e λ ⟩ ∣ ≤ C 2 − J β 0 ∥ ζ ∥ C β 0 , which proves (15).
Since β 1 < 0 \beta_1<0 β 1 < 0
sup λ ∥ e λ ∥ X 1 < ∞ . \sup_\lambda\|e_\lambda\|_{X_1}<\infty. λ sup ∥ e λ ∥ X 1 < ∞. Using 0 ≤ m J ≤ 1 0\le m_J\le1 0 ≤ m J ≤ 1
∥ Rec J ( c ) − Rec J ( c ′ ) ∥ X 1 ≤ ∑ λ ∈ Λ J ∣ m J ( λ ) ∣ ∣ c λ − c λ ′ ∣ ∥ e λ ∥ X 1 ≤ C ∑ λ ∈ Λ J ∣ c λ − c λ ′ ∣ ≤ C N J 1 / 2 ∥ c − c ′ ∥ ℓ 2 . \begin{aligned}
\|\operatorname{Rec}_J(c)-\operatorname{Rec}_J(c')\|_{X_1}\le\sum_{\lambda\in\Lambda_J}|m_J(\lambda)||c_\lambda-c'_\lambda|\|e_\lambda\|_{X_1} \\
&\le C\sum_{\lambda\in\Lambda_J}|c_\lambda-c'_\lambda| \\
&\le CN_J^{1/2}\|c-c'\|_{\ell^2}.
\end{aligned} ∥ Rec J ( c ) − Rec J ( c ′ ) ∥ X 1 ≤ λ ∈ Λ J ∑ ∣ m J ( λ ) ∣∣ c λ − c λ ′ ∣∥ e λ ∥ X 1 ≤ C λ ∈ Λ J ∑ ∣ c λ − c λ ′ ∣ ≤ C N J 1/2 ∥ c − c ′ ∥ ℓ 2 . This proves (16). □ \square □
Initial-data measurements and the reduced map
Let x 1 , … , x N 0 x_1,\ldots,x_{N_0} x 1 , … , x N 0 δ 0 \delta_0 δ 0 u ⃗ 0 = ( u 0 ( x i ) ) i ≤ N 0 \vec{u}_0=(u_0(x_i))_{i\le N_0} u 0 = ( u 0 ( x i ) ) i ≤ N 0 Rec 0 ( v ⃗ ) \operatorname{Rec}_0(\vec{v}) Rec 0 ( v )
Lemma 3.4. Let ( x i ) i < N 0 (x_i)_{i<N_0} ( x i ) i < N 0 T d \mathbb{T}^d T d δ 0 \delta_0 δ 0 Rec 0 \operatorname{Rec}_0 Rec 0 u 0 u_0 u 0 L 0 L_0 L 0
∥ u 0 − Rec 0 ( ( u 0 ( x i ) ) i ) ∥ ∞ ≤ C d L 0 δ 0 . (18) \lVert u_0-\operatorname{Rec}_0((u_0(x_i))_i)\rVert_\infty\leq C_dL_0\delta_0. \tag*{(18)} ∥ u 0 − Rec 0 (( u 0 ( x i ) ) i ) ∥ ∞ ≤ C d L 0 δ 0 . (18) For arbitrary nodal vectors v , v ′ v,v' v , v ′
∥ Rec 0 ( v ) − Rec 0 ( v ′ ) ∥ ∞ ≤ ∣ v − v ′ ∣ ∞ . (19) \lVert\operatorname{Rec}_0(v)-\operatorname{Rec}_0(v')\rVert_\infty\leq\lvert v-v'\rvert_{\infty}. \tag*{(19)} ∥ Rec 0 ( v ) − Rec 0 ( v ′ ) ∥ ∞ ≤ ∣ v − v ′ ∣ ∞ . (19) Moreover, N 0 ≳ δ 0 − d N_0\gtrsim\delta_0^{-d} N 0 ≳ δ 0 − d
Proof. On each grid cell, multilinear interpolation has the form
Rec 0 ( ( u 0 ( x i ) ) i ) ( x ) = ∑ ν ∈ { 0 , 1 } d w ν ( x ) u 0 ( x ν ) , \operatorname{Rec}_0((u_0(x_i))_i)(x)=\sum_{\nu\in\{0,1\}^d}w_\nu(x)u_0(x_\nu), Rec 0 (( u 0 ( x i ) ) i ) ( x ) = ν ∈ { 0 , 1 } d ∑ w ν ( x ) u 0 ( x ν ) , where the vertex weights are nonnegative and sum to one. Every vertex x ν x_\nu x ν d T d ( x , x ν ) ≤ d δ 0 d_{\mathbb{T}^d}(x,x_\nu)\leq\sqrt{d}\,\delta_0 d T d ( x , x ν ) ≤ d δ 0
∣ u 0 ( x ) − Rec 0 ( ( u 0 ( x i ) ) i ) ( x ) ∣ ≤ ∑ ν w ν ( x ) L 0 d T d ( x , x ν ) ≤ d L 0 δ 0 . \lvert u_0(x)-\operatorname{Rec}_0((u_0(x_i))_i)(x)\rvert\leq\sum_\nu w_\nu(x)L_0d_{\mathbb{T}^d}(x,x_\nu)\leq\sqrt{d}\,L_0\delta_0. ∣ u 0 ( x ) − Rec 0 (( u 0 ( x i ) ) i ) ( x )∣ ≤ ν ∑ w ν ( x ) L 0 d T d ( x , x ν ) ≤ d L 0 δ 0 . The same convex-combination formula applied to v − v ′ v-v' v − v ′ δ 0 − d \delta_0^{-d} δ 0 − d
For z = ( v ⃗ , c ) ∈ R N 0 × R N J z=(\vec{v},c)\in\mathbb{R}^{N_0}\times\mathbb{R}^{N_J} z = ( v , c ) ∈ R N 0 × R N J
Γ J ( z ) : = S ( Rec 0 ( v ⃗ ) , Rec J ( c ) ) ∈ Y . (20) \Gamma_J(z):=\mathcal{S}\bigl(\operatorname{Rec}_0(\vec{v}),\operatorname{Rec}_J(c)\bigr)\in Y. \tag*{(20)} Γ J ( z ) := S ( Rec 0 ( v ) , Rec J ( c ) ) ∈ Y . (20) Lemma 3.5. The map Γ J \Gamma_J Γ J
∥ Γ J ( v ⃗ , c ) − Γ J ( v ⃗ ′ , c ′ ) ∥ Y ≤ L J ( ∥ v ⃗ − v ⃗ ′ ∥ ∞ + ∥ c − c ′ ∥ 2 ) , \lVert\Gamma_J(\vec{v},c)-\Gamma_J(\vec{v}',c')\rVert_Y\leq L_J\bigl(\lVert\vec{v}-\vec{v}'\rVert_\infty+\lVert c-c'\rVert_2\bigr), ∥ Γ J ( v , c ) − Γ J ( v ′ , c ′ ) ∥ Y ≤ L J ( ∥ v − v ′ ∥ ∞ + ∥ c − c ′ ∥ 2 ) , where
L J : = C e L F T max { 1 , N J 1 / 2 } , L_J:=Ce^{L_FT}\max\{1,N_J^{1/2}\}, L J := C e L F T max { 1 , N J 1/2 } , and C C C J J J
Proof. By Theorem 2.5 , (18), and (16),
∥ Γ J ( v ⃗ , c ) − Γ J ( v ⃗ ′ , c ′ ) ∥ Y ≤ C e L F T ( ∥ v ⃗ − v ⃗ ′ ∥ ∞ + N J 1 / 2 ∥ c − c ′ ∥ 2 ) ≤ L J ( ∥ v ⃗ − v ⃗ ′ ∥ ∞ + ∥ c − c ′ ∥ 2 ) . \lVert\Gamma_J(\vec{v},c)-\Gamma_J(\vec{v}',c')\rVert_Y\leq Ce^{L_FT}\bigl(\lVert\vec{v}-\vec{v}'\rVert_\infty+N_J^{1/2}\lVert c-c'\rVert_2\bigr)
\leq L_J\bigl(\lVert\vec{v}-\vec{v}'\rVert_\infty+\lVert c-c'\rVert_2\bigr). ∥ Γ J ( v , c ) − Γ J ( v ′ , c ′ ) ∥ Y ≤ C e L F T ( ∥ v − v ′ ∥ ∞ + N J 1/2 ∥ c − c ′ ∥ 2 ) ≤ L J ( ∥ v − v ′ ∥ ∞ + ∥ c − c ′ ∥ 2 ) . □
Proposition 3.6. There is a constant C C C J J J δ 0 \delta_0 δ 0 u 0 ∈ G 0 u_0\in\mathcal{G}_0 u 0 ∈ G 0 ζ ∈ X 0 \zeta\in X_0 ζ ∈ X 0
∥ S ( u 0 , ζ ) − Γ J ( u ⃗ 0 , c J ( ζ ) ) ∥ Y ≤ C e L F T ( L 0 δ 0 + 2 − J δ ∥ ζ ∥ X 0 ) . (21) \lVert\mathcal{S}(u_0,\zeta)-\Gamma_J(\vec{u}_0,c_J(\zeta))\rVert_Y\leq Ce^{L_FT}\left(L_0\delta_0+2^{-J\delta}\lVert\zeta\rVert_{X_0}\right). \tag*{(21)} ∥ S ( u 0 , ζ ) − Γ J ( u 0 , c J ( ζ )) ∥ Y ≤ C e L F T ( L 0 δ 0 + 2 − J δ ∥ ζ ∥ X 0 ) . (21) Equivalently, in terms of the number of forcing measurements,
∥ S ( u 0 , ζ ) − Γ J ( u ⃗ 0 , c J ( ζ ) ) ∥ Y ≤ C e L F T ( L 0 δ 0 + N J − δ / ( d + 2 ) ∥ ζ ∥ X 0 ) . (22) \lVert\mathcal{S}(u_0,\zeta)-\Gamma_J(\vec{u}_0,c_J(\zeta))\rVert_Y\leq Ce^{L_FT}\left(L_0\delta_0+N_J^{-\delta/(d+2)}\lVert\zeta\rVert_{X_0}\right). \tag*{(22)} ∥ S ( u 0 , ζ ) − Γ J ( u 0 , c J ( ζ )) ∥ Y ≤ C e L F T ( L 0 δ 0 + N J − δ / ( d + 2 ) ∥ ζ ∥ X 0 ) . (22) Proof. By (10) and (19),
Γ J ( u ⃗ 0 , c J ( ζ ) ) = S ( Rec 0 ( u ⃗ 0 ) , Π J ζ ) . \Gamma_J(\vec{u}_0,c_J(\zeta))=\mathcal{S}(\operatorname{Rec}_0(\vec{u}_0),\Pi_J\zeta). Γ J ( u 0 , c J ( ζ )) = S ( Rec 0 ( u 0 ) , Π J ζ ) . The stability estimate (6) therefore bounds the left-hand side of (20) by a fixed multiple of
∥ u 0 − Rec 0 ( u ⃗ 0 ) ∥ ∞ + ∥ ( I − Π J ) ζ ∥ X 1 . \lVert u_0-\operatorname{Rec}_0(\vec{u}_0)\rVert_\infty+\lVert(I-\Pi_J)\zeta\rVert_{X_1}. ∥ u 0 − Rec 0 ( u 0 ) ∥ ∞ + ∥( I − Π J ) ζ ∥ X 1 . Apply (17) to the first term and Lemma 3.2 to the second. This proves (20). Finally, N J ≍ 2 J ( d + 2 ) N_J\asymp2^{J(d+2)} N J ≍ 2 J ( d + 2 ) 2 − J δ ≍ N J − δ / ( d + 2 ) 2^{-J\delta}\asymp N_J^{-\delta/(d+2)} 2 − J δ ≍ N J − δ / ( d + 2 )
Remark 3.7. The exponent δ = β 0 − β 1 \delta=\beta_0-\beta_1 δ = β 0 − β 1 (22) , but the stability space must still satisfy β 1 > − 2 \beta_1>-2 β 1 > − 2 δ < β 0 + 2 \delta<\beta_0+2 δ < β 0 + 2 ( β 0 + 2 ) / ( d + 2 ) (\beta_0+2)/(d+2) ( β 0 + 2 ) / ( d + 2 ) β 1 ↓ − 2 \beta_1\downarrow-2 β 1 ↓ − 2
Deep neural operator architecture
Let σ ( r ) = max { r , 0 } \sigma(r)=\max\{r,0\} σ ( r ) = max { r , 0 }
Definition 3.8 (ReLU network class). Let D ⊂ R n D\subset\mathbb{R}^n D ⊂ R n
Φ ∈ FNN ( n , m , L , p , K , κ , R ; D ) \Phi\in\operatorname{FNN}(n,m,L,p,K,\kappa,R;D) Φ ∈ FNN ( n , m , L , p , K , κ , R ; D ) if Φ : R n → R m \Phi:\mathbb{R}^n\to\mathbb{R}^m Φ : R n → R m
x 0 = x , x ℓ = σ ( W ℓ x ℓ − 1 + b ℓ ) , 1 ≤ ℓ < L , Φ ( x ) = W L x L − 1 + b L , \begin{aligned}
x_0&=x,\\
x_\ell&=\sigma(W_\ell x_{\ell-1}+b_\ell),\qquad1\le\ell<L,\\
\Phi(x)&=W_Lx_{L-1}+b_L,
\end{aligned} x 0 x ℓ Φ ( x ) = x , = σ ( W ℓ x ℓ − 1 + b ℓ ) , 1 ≤ ℓ < L , = W L x L − 1 + b L , such that every hidden layer has width at most p p p K K K κ \kappa κ ∥ Φ ∥ L ∞ ( D ) ≤ R \|\Phi\|_{L^\infty(D)}\le R ∥Φ ∥ L ∞ ( D ) ≤ R L L L p p p K K K κ \kappa κ
This is the network model used in [5 ] . We shall use the following elementary constructions.
Lemma 3.9. The following statements hold.
For a < b a<b a < b [ a , b ] [a,b] [ a , b ]
P [ a , b ] ( x ) = a + σ ( x − a ) − σ ( x − b ) . P_{[a,b]}(x)=a+\sigma(x-a)-\sigma(x-b). P [ a , b ] ( x ) = a + σ ( x − a ) − σ ( x − b ) . Applying this formula coordinatewise gives an exact ReLU realization of the projection onto a rectangular box.
For c ∈ R c\in\mathbb{R} c ∈ R r > 0 r>0 r > 0
H c , r ( x ) = σ ( 1 − r − 1 ( σ ( x − c ) + σ ( c − x ) ) ) = ( 1 − ∣ x − c ∣ r ) + H_{c,r}(x)=\sigma\left(1-r^{-1}\left(\sigma(x-c)+\sigma(c-x)\right)\right)=\left(1-\frac{\lvert x-c\rvert}{r}\right)_+ H c , r ( x ) = σ ( 1 − r − 1 ( σ ( x − c ) + σ ( c − x ) ) ) = ( 1 − r ∣ x − c ∣ ) + is represented exactly by a fixed-depth ReLU network.
Let M ≥ 1 M\ge1 M ≥ 1 τ ∈ ( 0 , 1 ) \tau\in(0,1) τ ∈ ( 0 , 1 ) Mult M , τ : R 2 → R \operatorname{Mult}_{M,\tau}:\mathbb{R}^2\to\mathbb{R} Mult M , τ : R 2 → R
sup ∣ x ∣ , ∣ y ∣ ≤ M ∣ Mult M , τ ( x , y ) − x y ∣ ≤ τ . \sup_{\lvert x\rvert,\lvert y\rvert\le M}\left\lvert\operatorname{Mult}_{M,\tau}(x,y)-xy\right\rvert\le\tau. ∣ x ∣ , ∣ y ∣ ≤ M sup ∣ Mult M , τ ( x , y ) − x y ∣ ≤ τ . Its width is bounded by an absolute constant, and its depth and number of nonzero parameters are O M ( log ( τ − 1 ) ) O_M(\log(\tau^{-1})) O M ( log ( τ − 1 ))
Consequently, for every integer q ≥ 2 q\ge2 q ≥ 2 q q q [ 0 , 1 ] [0,1] [ 0 , 1 ] τ \tau τ O ( log q log ( q / τ ) ) O(\log q\log(q/\tau)) O ( log q log ( q / τ )) O ( q log ( q / τ ) ) O(q\log(q/\tau)) O ( q log ( q / τ )) q q q O q ( log ( 1 / τ ) ) O_q(\log(1/\tau)) O q ( log ( 1/ τ ))
Proof. The first two statements follow from
σ ( x ) = max { x , 0 } , σ ( x − c ) + σ ( c − x ) = ∣ x − c ∣ . \sigma(x)=\max\{x,0\},\qquad\sigma(x-c)+\sigma(c-x)=\lvert x-c\rvert. σ ( x ) = max { x , 0 } , σ ( x − c ) + σ ( c − x ) = ∣ x − c ∣ . The third statement is [5 Lemma 3.1] .
For the product approximation, let
Q : = 2 ⌈ log 2 q ⌉ , q ≤ Q < 2 q , Q:=2^{\lceil\log_2 q\rceil},\qquad q\le Q<2q, Q := 2 ⌈ l o g 2 q ⌉ , q ≤ Q < 2 q , and pad the q q q Q − q Q-q Q − q 1 1 1 Q Q Q
( u , v ) ⟼ P [ 0 , 1 ] ( Mult 1 , δ ( u , v ) ) , δ : = τ 2 q . (u,v)\longmapsto P_{[0,1]}(\operatorname{Mult}_{1,\delta}(u,v)),\qquad\delta:=\frac{\tau}{2q}. ( u , v ) ⟼ P [ 0 , 1 ] ( Mult 1 , δ ( u , v )) , δ := 2 q τ . To estimate the error, let a , b ∈ [ 0 , 1 ] a,b\in[0,1] a , b ∈ [ 0 , 1 ] a ~ , b ~ ∈ [ 0 , 1 ] \tilde{a},\tilde{b}\in[0,1] a ~ , b ~ ∈ [ 0 , 1 ] P [ 0 , 1 ] P_{[0,1]} P [ 0 , 1 ] 1 1 1 a b ab ab
∣ P [ 0 , 1 ] ( Mult 1 , δ ( a ~ , b ~ ) ) − a b ∣ ≤ ∣ Mult 1 , δ ( a ~ , b ~ ) − a ~ b ~ ∣ + ∣ a ~ b ~ − a b ∣ ≤ δ + ∣ a ~ − a ∣ + ∣ b ~ − b ∣ . \left|P_{[0,1]}(\operatorname{Mult}_{1,\delta}(\tilde{a},\tilde{b}))-ab\right|
\leq\left|\operatorname{Mult}_{1,\delta}(\tilde{a},\tilde{b})-\tilde{a}\tilde{b}\right|+\left|\tilde{a}\tilde{b}-ab\right|
\leq\delta+|\tilde{a}-a|+|\tilde{b}-b|. P [ 0 , 1 ] ( Mult 1 , δ ( a ~ , b ~ )) − ab ≤ Mult 1 , δ ( a ~ , b ~ ) − a ~ b ~ + a ~ b ~ − ab ≤ δ + ∣ a ~ − a ∣ + ∣ b ~ − b ∣. The leaf errors vanish, so induction over the tree gives a root error of at most
( Q − 1 ) δ < 2 q δ = τ . (Q-1)\delta<2q\delta=\tau. ( Q − 1 ) δ < 2 q δ = τ . The tree has log 2 Q = O ( log q ) \log_2 Q=O(\log q) log 2 Q = O ( log q ) Q − 1 = O ( q ) Q-1=O(q) Q − 1 = O ( q ) O ( log ( q / τ ) ) O(\log(q/\tau)) O ( log ( q / τ )) [ 0 , 1 ] [0,1] [ 0 , 1 ]
depth = O ( log q log ( q / τ ) ) , size = O ( q log ( q / τ ) ) . \operatorname{depth}=O\bigl(\log q\log(q/\tau)\bigr),\qquad\operatorname{size}=O\bigl(q\log(q/\tau)\bigr). depth = O ( log q log ( q / τ ) ) , size = O ( q log ( q / τ ) ) . For fixed q q q O q ( log ( 1 / τ ) ) O_q(\log(1/\tau)) O q ( log ( 1/ τ ))
For fixed J J J δ 0 \delta_0 δ 0
E J , δ 0 ( u 0 , ζ ) : = ( u ⃗ 0 , c J ( ζ ) ) ∈ R N 0 + N J . (23) \mathcal{E}_{J,\delta_0}(u_0,\zeta):=(\vec{u}_0,c_J(\zeta))\in\mathbb{R}^{N_0+N_J}. \tag*{(23)} E J , δ 0 ( u 0 , ζ ) := ( u 0 , c J ( ζ )) ∈ R N 0 + N J . (23) The branch–trunk architecture considered below has the form
Γ θ ( E J , δ 0 ( u 0 , ζ ) ) ( y ) = ∑ ℓ = 1 N 1 a ~ ℓ ( u ⃗ 0 , c J ( ζ ) ; θ b ) q ~ ℓ ( y ; θ t ) , y = ( t , x ) ∈ D T . \Gamma_\theta\bigl(\mathcal{E}_{J,\delta_0}(u_0,\zeta)\bigr)(y)
=\sum_{\ell=1}^{N_1}\widetilde{a}_{\ell}(\vec{u}_0,c_J(\zeta);\theta_b)\widetilde{q}_{\ell}(y;\theta_t),
\qquad y=(t,x)\in D_T. Γ θ ( E J , δ 0 ( u 0 , ζ ) ) ( y ) = ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( ζ ) ; θ b ) q ℓ ( y ; θ t ) , y = ( t , x ) ∈ D T . The branch networks a ~ ℓ \widetilde{a}_{\ell} a ℓ q ~ ℓ \widetilde{q}_{\ell} q ℓ
A finite-dimensional partition
The partition of unity is applied only after the measurement map has placed the input in a finite-dimensional Euclidean space.
Lemma 3.10. Let K ⊂ R N K\subset\mathbb{R}^{N} K ⊂ R N Γ : K → Y \Gamma:K\to Y Γ : K → Y L L L z 1 , … , z m z^1,\ldots,z^m z 1 , … , z m r / 2 r/2 r /2 K K K
h k ( z ) : = ( 1 − ∣ z − z k ∣ 2 r ) + , ρ k ( z ) : = h k ( z ) ∑ j = 1 m h j ( z ) . h_k(z):=\left(1-\frac{|z-z^k|_2}{r}\right)_+,\qquad\rho_k(z):=\frac{h_k(z)}{\sum_{j=1}^{m}h_j(z)}. h k ( z ) := ( 1 − r ∣ z − z k ∣ 2 ) + , ρ k ( z ) := ∑ j = 1 m h j ( z ) h k ( z ) . Then the denominator is at least 1 / 2 1/2 1/2 K K K ρ k \rho_k ρ k
A r ( z ) : = ∑ k = 1 m ρ k ( z ) Γ ( z k ) ⟹ sup z ∈ K ∥ Γ ( z ) − A r ( z ) ∥ Y ≤ L r . A_r(z):=\sum_{k=1}^{m}\rho_k(z)\Gamma(z^k)\quad\Longrightarrow\quad\sup_{z\in K}\|\Gamma(z)-A_r(z)\|_Y\leq Lr. A r ( z ) := k = 1 ∑ m ρ k ( z ) Γ ( z k ) ⟹ z ∈ K sup ∥Γ ( z ) − A r ( z ) ∥ Y ≤ L r . Proof. Every z ∈ K z\in K z ∈ K r / 2 r/2 r /2 1 / 2 1/2 1/2 ρ k ( z ) > 0 \rho_k(z)>0 ρ k ( z ) > 0 ∣ z − z k ∣ 2 < r |z-z^k|_2<r ∣ z − z k ∣ 2 < r
∥ Γ ( z ) − A r ( z ) ∥ Y ≤ ∑ k ρ k ( z ) ∥ Γ ( z ) − Γ ( z k ) ∥ Y ≤ L r . \|\Gamma(z)-A_r(z)\|_Y\leq\sum_k\rho_k(z)\|\Gamma(z)-\Gamma(z^k)\|_Y\leq Lr. ∥Γ ( z ) − A r ( z ) ∥ Y ≤ k ∑ ρ k ( z ) ∥Γ ( z ) − Γ ( z k ) ∥ Y ≤ L r . Relation to Wiener chaos expansions
For the linear equation, the Fourier measurements determine a finite sum in the first Wiener chaos. The following result also gives an exact covariance formula for the mean-square truncation error.
Lemma 3.11. Assume F ≡ 0 F \equiv0 F ≡ 0 e λ e_\lambda e λ
ξ λ : = ⟨ η , e λ ⟩ , z λ : = K e λ , \xi_\lambda:= \langle\eta,e_\lambda\rangle,\qquad z_\lambda:= Ke_\lambda, ξ λ := ⟨ η , e λ ⟩ , z λ := K e λ , and write Σ λ μ : = E [ ξ λ ξ μ ] \Sigma_{\lambda\mu} := \mathbb{E}[\xi_\lambda\xi_\mu] Σ λ μ := E [ ξ λ ξ μ ]
For every J J J
S ( u 0 , Π J η ) = e t Δ u 0 + ∑ λ ∈ Λ J m J ( λ ) ξ λ z λ . (24) S(u_0,\Pi_J\eta)=e^{t\Delta}u_0+\sum_{\lambda\in\Lambda_J}m_J(\lambda)\xi_\lambda z_\lambda. \tag*{(24)} S ( u 0 , Π J η ) = e t Δ u 0 + λ ∈ Λ J ∑ m J ( λ ) ξ λ z λ . (24) The solutions in (24) converge to S ( u 0 , η ) S(u_0,\eta) S ( u 0 , η ) L 2 ( Ω ; C ( D T ) ) L^2(\Omega;C(D_T)) L 2 ( Ω ; C ( D T )) K η K\eta K η ( ξ λ ) λ (\xi_\lambda)_\lambda ( ξ λ ) λ
The truncation error has the exact covariance representation
E ∥ S ( u 0 , η ) − S ( u 0 , Π J η ) ∥ L 2 ( D T ) 2 = lim L → ∞ ∑ λ , μ ∈ Λ L ( m L ( λ ) − m J ( λ ) ) ( m L ( μ ) − m J ( μ ) ) Σ λ μ ⟨ z λ , z μ ⟩ L 2 ( D T ) . (25) \mathbb{E}\left\|S(u_0,\eta)-S(u_0,\Pi_J\eta)\right\|_{L^2(D_T)}^2
=
\lim_{L\to\infty}\sum_{\lambda,\mu\in\Lambda_L}
\left(m_L(\lambda)-m_J(\lambda)\right)
\left(m_L(\mu)-m_J(\mu)\right)
\Sigma_{\lambda\mu}\langle z_\lambda,z_\mu\rangle_{L^2(D_T)}. \tag*{(25)} E ∥ S ( u 0 , η ) − S ( u 0 , Π J η ) ∥ L 2 ( D T ) 2 = L → ∞ lim λ , μ ∈ Λ L ∑ ( m L ( λ ) − m J ( λ ) ) ( m L ( μ ) − m J ( μ ) ) Σ λ μ ⟨ z λ , z μ ⟩ L 2 ( D T ) . (25) each inner sum being finite and exact for the smooth outer cutoff Π L \Pi_L Π L L > J L>J L > J Σ λ μ = σ λ 2 1 { λ = μ } \Sigma_{\lambda\mu}=\sigma_\lambda^2\mathbf{1}_{\{\lambda=\mu\}} Σ λ μ = σ λ 2 1 { λ = μ }
E ∥ S ( u 0 , η ) − S ( u 0 , Π J η ) ∥ L 2 ( D T ) 2 = ∑ λ ( 1 − m J ( λ ) ) 2 σ λ 2 ∥ z λ ∥ L 2 ( D T ) 2 . (26) \mathbb{E}\left\|S(u_0,\eta)-S(u_0,\Pi_J\eta)\right\|_{L^2(D_T)}^2
=
\sum_\lambda\left(1-m_J(\lambda)\right)^2\sigma_\lambda^2\|z_\lambda\|_{L^2(D_T)}^2. \tag*{(26)} E ∥ S ( u 0 , η ) − S ( u 0 , Π J η ) ∥ L 2 ( D T ) 2 = λ ∑ ( 1 − m J ( λ ) ) 2 σ λ 2 ∥ z λ ∥ L 2 ( D T ) 2 . (26) Proof. For F ≡ 0 F\equiv0 F ≡ 0
S ( u 0 , ϕ ) = e t Δ u 0 + K ϕ . S(u_0,\phi)=e^{t\Delta}u_0+K\phi. S ( u 0 , ϕ ) = e t Δ u 0 + K ϕ . Since
Π J η = ∑ λ ∈ Λ J m J ( λ ) ξ λ e λ , \Pi_J\eta=\sum_{\lambda\in\Lambda_J}m_J(\lambda)\xi_\lambda e_\lambda, Π J η = λ ∈ Λ J ∑ m J ( λ ) ξ λ e λ , linearity of K K K
S ( u 0 , Π J η ) = e t Δ u 0 + ∑ λ ∈ Λ J m J ( λ ) ξ λ z λ , S(u_0,\Pi_J\eta)=e^{t\Delta}u_0+\sum_{\lambda\in\Lambda_J}m_J(\lambda)\xi_\lambda z_\lambda, S ( u 0 , Π J η ) = e t Δ u 0 + λ ∈ Λ J ∑ m J ( λ ) ξ λ z λ , proving (24) .
Theorem 2.5 with L F = 0 L_F=0 L F = 0 Lemma 3.2 yield
∥ S ( u 0 , η ) − S ( u 0 , Π J η ) ∥ Y ≤ C 2 − J δ ∥ η ∥ X 0 . (27) \left\|S(u_0,\eta)-S(u_0,\Pi_J\eta)\right\|_Y\le C2^{-J\delta}\|\eta\|_{X_0}. \tag*{(27)} ∥ S ( u 0 , η ) − S ( u 0 , Π J η ) ∥ Y ≤ C 2 − J δ ∥ η ∥ X 0 . (27) By Lemma 2.3 ,
∥ η ∥ X 0 < ∞ almost surely , E ∥ η ∥ X 0 2 < ∞ . \|\eta\|_{X_0}<\infty\quad\text{almost surely},\qquad\mathbb{E}\|\eta\|_{X_0}^2<\infty. ∥ η ∥ X 0 < ∞ almost surely , E ∥ η ∥ X 0 2 < ∞. Thus (27) implies convergence both almost surely and in L 2 ( Ω ; C ( D T ) ) L^2(\Omega;C(D_T)) L 2 ( Ω ; C ( D T ))
K Π J η = ∑ λ ∈ Λ J m J ( λ ) ξ λ z λ K\Pi_J\eta=\sum_{\lambda\in\Lambda_J}m_J(\lambda)\xi_\lambda z_\lambda K Π J η = λ ∈ Λ J ∑ m J ( λ ) ξ λ z λ is a finite linear combination of the centered Gaussian coordinates ξ λ \xi_\lambda ξ λ z λ z_\lambda z λ L 2 L^2 L 2 K η K\eta K η e t Δ u 0 e^{t\Delta}u_0 e t Δ u 0 K η K\eta K η
Next fix J J J L > J L > J L > J Λ J ⊆ Λ L \Lambda_J \subseteq\Lambda_L Λ J ⊆ Λ L
K ( Π L − Π J ) η = ∑ λ ∈ Λ L ( m L ( λ ) − m J ( λ ) ) ξ λ z λ . K(\Pi_L-\Pi_J)\eta=\sum_{\lambda\in\Lambda_L}(m_L(\lambda)-m_J(\lambda))\xi_\lambda z_\lambda. K ( Π L − Π J ) η = λ ∈ Λ L ∑ ( m L ( λ ) − m J ( λ )) ξ λ z λ . Expanding the squared norm and taking expectations gives
E ∥ K ( Π L − Π J ) η ∥ L 2 ( D T ) 2 = ∑ λ , μ ∈ Λ L ( m L ( λ ) − m J ( λ ) ) ( m L ( μ ) − m J ( μ ) ) Σ λ μ ⟨ z λ , z μ ⟩ L 2 ( D T ) . \mathbb{E}\left\|K(\Pi_L-\Pi_J)\eta\right\|_{L^2(D_T)}^2
=
\sum_{\lambda,\mu\in\Lambda_L}(m_L(\lambda)-m_J(\lambda))(m_L(\mu)-m_J(\mu))\Sigma_{\lambda\mu}\langle z_\lambda,z_\mu\rangle_{L^2(D_T)}. E ∥ K ( Π L − Π J ) η ∥ L 2 ( D T ) 2 = λ , μ ∈ Λ L ∑ ( m L ( λ ) − m J ( λ )) ( m L ( μ ) − m J ( μ )) Σ λ μ ⟨ z λ , z μ ⟩ L 2 ( D T ) . Moreover, (27) implies
K ( Π L − Π J ) η ⟶ K ( I − Π J ) η in L 2 ( Ω ; L 2 ( D T ) ) . K(\Pi_L-\Pi_J)\eta\longrightarrow K(I-\Pi_J)\eta\quad\text{in }L^2(\Omega;L^2(D_T)). K ( Π L − Π J ) η ⟶ K ( I − Π J ) η in L 2 ( Ω ; L 2 ( D T )) . The second moments therefore converge, proving (25) .
If Σ \Sigma Σ
E ∥ K ( Π L − Π J ) η ∥ L 2 ( D T ) 2 = ∑ λ ∈ Λ L ( m L ( λ ) − m J ( λ ) ) 2 σ λ 2 ∥ z λ ∥ L 2 ( D T ) 2 . \mathbb{E}\left\|K(\Pi_L-\Pi_J)\eta\right\|_{L^2(D_T)}^2
=
\sum_{\lambda\in\Lambda_L}(m_L(\lambda)-m_J(\lambda))^2\sigma_\lambda^2\left\|z_\lambda\right\|_{L^2(D_T)}^2. E ∥ K ( Π L − Π J ) η ∥ L 2 ( D T ) 2 = λ ∈ Λ L ∑ ( m L ( λ ) − m J ( λ ) ) 2 σ λ 2 ∥ z λ ∥ L 2 ( D T ) 2 . For each λ \lambda λ (10) gives
( m L ( λ ) − m J ( λ ) ) 2 ↑ ( 1 − m J ( λ ) ) 2 as L → ∞ . (m_L(\lambda)-m_J(\lambda))^2 \uparrow(1-m_J(\lambda))^2\quad\text{as }L\to\infty. ( m L ( λ ) − m J ( λ ) ) 2 ↑ ( 1 − m J ( λ ) ) 2 as L → ∞. Extending each finite sum by zero outside Λ L \Lambda_L Λ L (26) . □
Approximation results
Function approximation: the trunk network
We first approximate functions v ( t , x ) v(t,x) v ( t , x ) D T = [ 0 , T ] × T d D_T=[0,T]\times\mathbb{T}^d D T = [ 0 , T ] × T d ∣ t − s ∣ 1 / 2 + d T d ( x , y ) |t-s|^{1/2}+d_{\mathbb{T}^d}(x,y) ∣ t − s ∣ 1/2 + d T d ( x , y ) h h h h 2 h^2 h 2 h h h
Theorem 4.1 (Parabolic function approximation). Let B , Λ ≥ 1 B,\Lambda\ge1 B , Λ ≥ 1 0 < γ ≤ 1 0<\gamma\le1 0 < γ ≤ 1 V \mathcal{V} V v ∈ C ( D T ) v\in C(D_T) v ∈ C ( D T )
∥ v ∥ ∞ ≤ B , ∣ v ( z ) − v ( z ′ ) ∣ ≤ Λ d s ( z , z ′ ) γ . \|v\|_\infty\le B,\qquad|v(z)-v(z')|\le\Lambda d_s(z,z')^\gamma. ∥ v ∥ ∞ ≤ B , ∣ v ( z ) − v ( z ′ ) ∣ ≤ Λ d s ( z , z ′ ) γ . For every ε T ∈ ( 0 , B ] \varepsilon_T\in(0,B] ε T ∈ ( 0 , B ] ε T ≤ 2 Λ \varepsilon_T\le2\Lambda ε T ≤ 2Λ z 1 , … , z N 1 ∈ D T z_1,\ldots,z_{N_1}\in D_T z 1 , … , z N 1 ∈ D T q ~ 1 , … , q ~ N 1 : R 1 + d → R \widetilde{q}_1,\ldots,\widetilde{q}_{N_1}:\mathbb{R}^{1+d}\to\mathbb{R} q 1 , … , q N 1 : R 1 + d → R
sup v ∈ V sup z ∈ D T ∣ v ( z ) − ∑ ℓ = 1 N 1 v ( z ℓ ) q ~ ℓ ( z ) ∣ ≤ ε T . (28) \sup_{v\in\mathcal{V}}\sup_{z\in D_T}\left|v(z)-\sum_{\ell=1}^{N_1}v(z_\ell)\widetilde{q}_\ell(z)\right|\le\varepsilon_T. \tag*{(28)} v ∈ V sup z ∈ D T sup v ( z ) − ℓ = 1 ∑ N 1 v ( z ℓ ) q ℓ ( z ) ≤ ε T . (28) sup z ∈ D T ∑ ℓ = 1 N 1 ∣ q ~ ℓ ( z ) ∣ ≤ 3 2 . (29) \sup_{z\in D_T}\sum_{\ell=1}^{N_1}|\widetilde{q}_\ell(z)|\le\frac{3}{2}. \tag*{(29)} z ∈ D T sup ℓ = 1 ∑ N 1 ∣ q ℓ ( z ) ∣ ≤ 2 3 . (29) Moreover,
N 1 ≍ ( Λ ε T ) ( d + 2 ) / γ . (30) N_1\asymp\left(\frac{\Lambda}{\varepsilon_T}\right)^{(d+2)/\gamma}. \tag*{(30)} N 1 ≍ ( ε T Λ ) ( d + 2 ) / γ . (30) Each trunk network may be chosen with depth and number of nonzero parameters O ( ( d + 1 ) log ( B N 1 / ε T ) ) O((d+1)\log(BN_1/\varepsilon_T)) O (( d + 1 ) log ( B N 1 / ε T )) d d d
Proof. We first construct an interpolant from finitely many point values, then approximate its basis functions by ReLU networks. Choose
h : = 1 1 + d ( ε T 2 Λ ) 1 / γ . h := \frac{1}{1+\sqrt{d}}\left(\frac{\varepsilon_T}{2\Lambda}\right)^{1/\gamma}. h := 1 + d 1 ( 2Λ ε T ) 1/ γ . Since ε T ≤ 2 Λ \varepsilon_T \le2\Lambda ε T ≤ 2Λ h < 1 h<1 h < 1
n t : = ⌈ T h 2 ⌉ , n x : = ⌈ 2 π h ⌉ , τ : = T n t , σ : = 2 π n x . n_t := \left\lceil\frac{T}{h^2}\right\rceil,\qquad n_x := \left\lceil\frac{2\pi}{h}\right\rceil,\qquad\tau:= \frac{T}{n_t},\qquad\sigma:= \frac{2\pi}{n_x}. n t := ⌈ h 2 T ⌉ , n x := ⌈ h 2 π ⌉ , τ := n t T , σ := n x 2 π . The temporal grid t i = i τ t_i=i\tau t i = i τ 0 ≤ i ≤ n t 0\le i\le n_t 0 ≤ i ≤ n t τ ≤ h 2 \tau\le h^2 τ ≤ h 2 x j = j σ x_j=j\sigma x j = j σ 0 ≤ j < n x 0\le j<n_x 0 ≤ j < n x σ ≤ h \sigma\le h σ ≤ h
For 0 ≤ i ≤ n t 0\le i\le n_t 0 ≤ i ≤ n t ϑ i \vartheta_i ϑ i [ 0 , T ] [0,T] [ 0 , T ]
ϑ i ( t k ) = δ i k , 0 ≤ k ≤ n t . \vartheta_i(t_k)=\delta_{ik},\qquad0\le k\le n_t. ϑ i ( t k ) = δ ik , 0 ≤ k ≤ n t . and linear on each interval [ t k , t k + 1 ] [t_k,t_{k+1}] [ t k , t k + 1 ] 0 ≤ j < n x 0\le j<n_x 0 ≤ j < n x ψ j \psi_j ψ j
ψ j ( x k ) = δ j k , 0 ≤ k < n x . \psi_j(x_k)=\delta_{jk},\qquad0\le k<n_x. ψ j ( x k ) = δ j k , 0 ≤ k < n x . Here each function equals one at its own grid point and zero at all the other grid points. The spatial functions are linear on each grid interval, including [ x n x − 1 , 2 π ] [x_{n_x-1},2\pi] [ x n x − 1 , 2 π ] 2 π 2\pi 2 π
ϑ i , ψ j ≥ 0 , ∑ i = 0 n t ϑ i ( t ) = 1 , ∑ j = 0 n x − 1 ψ j ( x ) = 1. \vartheta_i,\psi_j\ge0,\qquad\sum_{i=0}^{n_t}\vartheta_i(t)=1,\qquad\sum_{j=0}^{n_x-1}\psi_j(x)=1. ϑ i , ψ j ≥ 0 , i = 0 ∑ n t ϑ i ( t ) = 1 , j = 0 ∑ n x − 1 ψ j ( x ) = 1. For α = ( i , j 1 , … , j d ) \alpha=(i,j_1,\ldots,j_d) α = ( i , j 1 , … , j d )
z α : = ( t i , x j 1 , … , x j d ) , φ α ( t , x ) : = ϑ i ( t ) ∏ q = 1 d ψ j q ( x q ) . z_\alpha:=(t_i,x_{j_1},\ldots,x_{j_d}),\qquad\varphi_\alpha(t,x):=\vartheta_i(t)\prod_{q=1}^{d}\psi_{j_q}(x_q). z α := ( t i , x j 1 , … , x j d ) , φ α ( t , x ) := ϑ i ( t ) q = 1 ∏ d ψ j q ( x q ) . Then φ α ≥ 0 \varphi_\alpha\ge0 φ α ≥ 0 ∑ α φ α = 1 \sum_\alpha\varphi_\alpha=1 ∑ α φ α = 1 φ α \varphi_\alpha φ α z α z_\alpha z α τ \tau τ d σ \sqrt{d}\sigma d σ
d s ( z , z α ) ≤ τ + d σ ≤ ( 1 + d ) h . d_s(z,z_\alpha)\le\sqrt{\tau}+\sqrt{d}\sigma\le(1+\sqrt{d})h. d s ( z , z α ) ≤ τ + d σ ≤ ( 1 + d ) h . For v ∈ V v\in\mathcal{V} v ∈ V
I h v ( z ) : = ∑ α v ( z α ) φ α ( z ) . I_hv(z):=\sum_\alpha v(z_\alpha)\varphi_\alpha(z). I h v ( z ) := α ∑ v ( z α ) φ α ( z ) . The partition-of-unity property and the assumed Hölder bound give
∣ v ( z ) − I h v ( z ) ∣ ≤ ∑ α : φ α ( z ) > 0 φ α ( z ) ∣ v ( z ) − v ( z α ) ∣ ≤ Λ ( ( 1 + d ) h ) γ = ε T 2 . \begin{aligned}
|v(z)-I_hv(z)|
&\le\sum_{\alpha:\,\varphi_\alpha(z)>0}\varphi_\alpha(z)|v(z)-v(z_\alpha)|\\
&\le\Lambda((1+\sqrt{d})h)^\gamma=\frac{\varepsilon_T}{2}.
\end{aligned} ∣ v ( z ) − I h v ( z ) ∣ ≤ α : φ α ( z ) > 0 ∑ φ α ( z ) ∣ v ( z ) − v ( z α ) ∣ ≤ Λ (( 1 + d ) h ) γ = 2 ε T . This estimate is uniform over v ∈ V v\in\mathcal{V} v ∈ V z ∈ D T z\in D_T z ∈ D T
N 1 = ( n t + 1 ) n x d . N_1=(n_t+1)n_x^d. N 1 = ( n t + 1 ) n x d . Since h < 1 h<1 h < 1 n t + 1 ≍ h − 2 n_t+1\asymp h^{-2} n t + 1 ≍ h − 2 n x ≍ h − 1 n_x\asymp h^{-1} n x ≍ h − 1
N 1 ≍ h − ( d + 2 ) ≲ ( Λ ε T ) ( d + 2 ) / γ , N_1\asymp h^{-(d+2)}\lesssim\left(\frac{\Lambda}{\varepsilon_T}\right)^{(d+2)/\gamma}, N 1 ≍ h − ( d + 2 ) ≲ ( ε T Λ ) ( d + 2 ) / γ , which proves (29). The comparison constants depend only on T T T d d d γ \gamma γ
We next approximate the interpolation basis functions by ReLU networks. The temporal interpolation functions have the explicit form
ϑ i ( t ) = ( 1 − ∣ t − t i ∣ τ ) + , t ∈ [ 0 , T ] . \vartheta_i(t)=\left(1-\frac{|t-t_i|}{\tau}\right)_+,\qquad t\in[0,T]. ϑ i ( t ) = ( 1 − τ ∣ t − t i ∣ ) + , t ∈ [ 0 , T ] . On the coordinate interval [ 0 , 2 π ] [0,2\pi] [ 0 , 2 π ]
ψ j ( x ) = ∑ k = − 1 1 ( 1 − ∣ x − x j − 2 π k ∣ σ ) + . \psi_j(x)=\sum_{k=-1}^{1}\left(1-\frac{|x-x_j-2\pi k|}{\sigma}\right)_+. ψ j ( x ) = k = − 1 ∑ 1 ( 1 − σ ∣ x − x j − 2 π k ∣ ) + . By Lemma 3.9(2), each of these one-dimensional interpolation functions has an exact ReLU representation with fixed depth and a fixed number of nonzero parameters. Their values lie in [ 0 , 1 ] [0,1] [ 0 , 1 ] φ α \varphi_\alpha φ α d + 1 d+1 d + 1 δ T ∈ ( 0 , 1 ) \delta_T\in(0,1) δ T ∈ ( 0 , 1 ) q ~ α \widetilde{q}_\alpha q α
∥ φ α − q ~ α ∥ C ( D T ) ≤ δ T . \|\varphi_\alpha-\widetilde{q}_\alpha\|_{C(D_T)}\leq\delta_T. ∥ φ α − q α ∥ C ( D T ) ≤ δ T . Since ψ j ( 0 ) = ψ j ( 2 π ) \psi_j(0)=\psi_j(2\pi) ψ j ( 0 ) = ψ j ( 2 π ) j j j x q = 0 x_q=0 x q = 0 x q = 2 π x_q=2\pi x q = 2 π q ~ α \widetilde{q}_\alpha q α T d \mathbb{T}^d T d D T D_T D T ℓ = 1 , … , N 1 \ell=1,\ldots,N_1 ℓ = 1 , … , N 1 v ∈ V v\in\mathcal{V} v ∈ V ∣ v ( z ℓ ) ∣ ≤ B |v(z_\ell)|\leq B ∣ v ( z ℓ ) ∣ ≤ B
∥ v − ∑ ℓ = 1 N 1 v ( z ℓ ) q ~ ℓ ∥ C ( D T ) ≤ ∥ v − I h v ∥ C ( D T ) + ∑ ℓ = 1 N 1 ∣ v ( z ℓ ) ∣ ∥ φ ℓ − q ~ ℓ ∥ C ( D T ) ≤ ε T 2 + B N 1 δ T . \begin{aligned}
\left\|v-\sum_{\ell=1}^{N_1}v(z_\ell)\widetilde{q}_\ell\right\|_{C(D_T)}
\leq\|v-I_hv\|_{C(D_T)}+\sum_{\ell=1}^{N_1}|v(z_\ell)|\|\varphi_\ell-\widetilde{q}_\ell\|_{C(D_T)} \\
&\leq\frac{\varepsilon_T}{2}+BN_1\delta_T.
\end{aligned} v − ℓ = 1 ∑ N 1 v ( z ℓ ) q ℓ C ( D T ) ≤ ∥ v − I h v ∥ C ( D T ) + ℓ = 1 ∑ N 1 ∣ v ( z ℓ ) ∣∥ φ ℓ − q ℓ ∥ C ( D T ) ≤ 2 ε T + B N 1 δ T . Choose
δ T : = ε T 2 B N 1 . (31) \delta_T:=\frac{\varepsilon_T}{2BN_1}. \tag*{(31)} δ T := 2 B N 1 ε T . (31) Since ε T ≤ B \varepsilon_T\leq B ε T ≤ B ( 0 , 1 ) (0,1) ( 0 , 1 ) ε T \varepsilon_T ε T z ∈ D T z\in D_T z ∈ D T
∑ ℓ = 1 N 1 ∣ q ~ ℓ ( z ) ∣ ≤ ∑ ℓ = 1 N 1 φ ℓ ( z ) + ∑ ℓ = 1 N 1 ∣ q ~ ℓ ( z ) − φ ℓ ( z ) ∣ \sum_{\ell=1}^{N_1}|\widetilde{q}_\ell(z)|
\leq\sum_{\ell=1}^{N_1}\varphi_\ell(z)+\sum_{\ell=1}^{N_1}|\widetilde{q}_\ell(z)-\varphi_\ell(z)| ℓ = 1 ∑ N 1 ∣ q ℓ ( z ) ∣ ≤ ℓ = 1 ∑ N 1 φ ℓ ( z ) + ℓ = 1 ∑ N 1 ∣ q ℓ ( z ) − φ ℓ ( z ) ∣ ≤ 1 + N 1 δ T = 1 + ε T 2 B ≤ 3 2 . \leq1+N_1\delta_T=1+\frac{\varepsilon_T}{2B}\leq\frac{3}{2}. ≤ 1 + N 1 δ T = 1 + 2 B ε T ≤ 2 3 . This proves (28).
Finally, Lemma 3.9 gives each product network depth and number of nonzero parameters bounded by
O ( ( d + 1 ) log d + 1 δ T ) . O\left((d+1)\log\frac{d+1}{\delta_T}\right). O ( ( d + 1 ) log δ T d + 1 ) . Combining this with the exact representations of the interpolation functions and substituting (30) gives O ( ( d + 1 ) log ( B N 1 / ε T ) ) O((d+1)\log(BN_1/\varepsilon_T)) O (( d + 1 ) log ( B N 1 / ε T )) d d d
Functional approximation: the branch network
For y ∈ D T y \in D_T y ∈ D T
F y ( u 0 , ζ ) : = S ( u 0 , ζ ) ( y ) . \mathcal{F}_y(u_0,\zeta) := S(u_0,\zeta)(y). F y ( u 0 , ζ ) := S ( u 0 , ζ ) ( y ) . The forcing ζ \zeta ζ c J ( ζ ) c_J(\zeta) c J ( ζ ) (23) , we approximate the solution value F y ( u 0 , ζ ) \mathcal{F}_y(u_0,\zeta) F y ( u 0 , ζ ) ( u ⃗ 0 , c J ( ζ ) ) (\vec{u}_0,c_J(\zeta)) ( u 0 , c J ( ζ )) y y y Γ J ( z k ) ( y ) \Gamma_J(z^k)(y) Γ J ( z k ) ( y )
Fix R ≥ 1 R \ge1 R ≥ 1 Assumption 2.6 gives ∣ u 0 ( x i ) ∣ ≤ M \lvert u_0(x_i)\rvert\le M ∣ u 0 ( x i )∣ ≤ M { ∥ η ∥ X 0 ≤ R } \{\lVert\eta\rVert_{X_0} \le R\} {∥ η ∥ X 0 ≤ R } Lemma 3.3 gives
∥ c J ( η ) ∥ ℓ ∞ ≤ C 2 − J β 0 R . \lVert c_J(\eta)\rVert_{\ell^\infty} \le C2^{-J\beta_0}R. ∥ c J ( η ) ∥ ℓ ∞ ≤ C 2 − J β 0 R . We therefore set R J = C 2 − J β 0 R R_J = C2^{-J\beta_0}R R J = C 2 − J β 0 R
K J , R : = [ − M , M ] N 0 × [ − R J , R J ] N J . K_{J,R} := [-M,M]^{N_0} \times[-R_J,R_J]^{N_J}. K J , R := [ − M , M ] N 0 × [ − R J , R J ] N J . Then ( u ⃗ 0 , c J ( η ) ) ∈ K J , R (\vec{u}_0,c_J(\eta)) \in K_{J,R} ( u 0 , c J ( η )) ∈ K J , R K J , R K_{J,R} K J , R
We combine the reconstruction estimates with a partition of unity on K J , R K_{J,R} K J , R
Theorem 4.2. Let F : R → R F:\mathbb{R}\to\mathbb{R} F : R → R S S S Theorem 2.5 . Suppose that Assumptions 2.2 and 2.6 hold, and fix β 1 \beta_1 β 1 δ \delta δ (2) . Let R ≥ 1 R \ge1 R ≥ 1 ε B ∈ ( 0 , 1 ) \varepsilon_B \in(0,1) ε B ∈ ( 0 , 1 )
2 J ≤ C ( R ε B ) 1 / δ , δ 0 = c 0 ε B , r = ε B 8 L J , (32) 2^J \le C\left(\frac{R}{\varepsilon_B}\right)^{1/\delta}, \qquad\delta_0=c_0\varepsilon_B, \qquad r=\frac{\varepsilon_B}{8L_J}, \tag*{(32)} 2 J ≤ C ( ε B R ) 1/ δ , δ 0 = c 0 ε B , r = 8 L J ε B , (32) where c 0 > 0 c_0>0 c 0 > 0 d d d T T T L F L_F L F L 0 L_0 L 0 r / 2 r/2 r /2 ( z k ) k ≤ m (z^k)_{k\le m} ( z k ) k ≤ m K J , R K_{J,R} K J , R ρ ~ k : R N 0 + N J → R \widetilde{\rho}_k:\mathbb{R}^{N_0+N_J}\to\mathbb{R} ρ k : R N 0 + N J → R
sup u 0 ∈ G 0 sup ∥ ζ ∥ X 0 ≤ R sup z ∙ ∈ D T ∣ S ( u 0 , ζ ) ( z ∙ ) − ∑ k = 1 m Γ J ( z k ) ( z ∙ ) ρ ~ k ( u ⃗ 0 , c J ( ζ ) ) ∣ ≤ ε B . (33) \sup_{u_0\in\mathcal{G}_0}\sup_{\lVert\zeta\rVert_{X_0}\le R}\sup_{z_\bullet\in D_T}
\left\lvert S(u_0,\zeta)(z_\bullet)-\sum_{k=1}^{m}\Gamma_J(z^k)(z_\bullet)\widetilde{\rho}_k(\vec{u}_0,c_J(\zeta))\right\rvert
\le\varepsilon_B. \tag*{(33)} u 0 ∈ G 0 sup ∥ ζ ∥ X 0 ≤ R sup z ∙ ∈ D T sup S ( u 0 , ζ ) ( z ∙ ) − k = 1 ∑ m Γ J ( z k ) ( z ∙ ) ρ k ( u 0 , c J ( ζ )) ≤ ε B . (33) The networks can be chosen so that
∑ k = 1 m ∣ ρ ~ k ( z ) ∣ ≤ 2 , z ∈ R N 0 + N J . (34) \sum_{k=1}^{m}\lvert\widetilde{\rho}_k(z)\rvert\le2,\qquad z\in\mathbb{R}^{N_0+N_J}. \tag*{(34)} k = 1 ∑ m ∣ ρ k ( z )∣ ≤ 2 , z ∈ R N 0 + N J . (34) Moreover, the number m m m
m ≤ ( 1 + 2 ( M ∨ R J ) N r ) N , N : = N 0 + N J . (35) m\le\left(1+\frac{2(M\vee R_J)\sqrt{N}}{r}\right)^N,\qquad N:=N_0+N_J. \tag*{(35)} m ≤ ( 1 + r 2 ( M ∨ R J ) N ) N , N := N 0 + N J . (35) Choose a constant C B ≥ 1 C_B\ge1 C B ≥ 1 d d d T T T β 0 \beta_0 β 0 β 1 \beta_1 β 1 M M M L 0 L_0 L 0 L F L_F L F ∣ F ( 0 ) ∣ \lvert F(0)\rvert ∣ F ( 0 )∣ B J : = C B ( 1 + N J R J ) B_J:=C_B(1+N_JR_J) B J := C B ( 1 + N J R J ) Γ J ( z k ) \Gamma_J(z^k) Γ J ( z k ) Y Y Y
H B : = log ( e N m ( B J + 1 ) ε B ) . H_B:=\log\left(\frac{eNm(B_J+1)}{\varepsilon_B}\right). H B := log ( ε B e N m ( B J + 1 ) ) . Each ρ ~ k \widetilde{\rho}_k ρ k C log ( e N ) H B C\log(eN)H_B C log ( e N ) H B C N H B CNH_B C N H B ρ ~ : = ( ρ ~ 1 , … , ρ ~ m ) \widetilde{\rho}:=(\widetilde{\rho}_1,\ldots,\widetilde{\rho}_m) ρ := ( ρ 1 , … , ρ m )
L ρ ≤ C log ( e N ) H B , S ρ ≤ C m N H B . (36) L_\rho\le C\log(eN)H_B,\qquad S_\rho\le CmNH_B. \tag*{(36)} L ρ ≤ C log ( e N ) H B , S ρ ≤ C m N H B . (36) where L ρ L_\rho L ρ S ρ S_\rho S ρ K J , R K_{J,R} K J , R C C C J J J R R R ε B \varepsilon_B ε B N N N m m m
Proof. Fix u 0 ∈ G 0 u_0 \in\mathcal{G}_0 u 0 ∈ G 0 ζ ∈ X 0 \zeta\in X_0 ζ ∈ X 0 ∥ ζ ∥ X 0 ≤ R \lVert\zeta\rVert_{X_0} \le R ∥ ζ ∥ X 0 ≤ R u ^ 0 : = Rec 0 ( u ⃗ 0 ) \widehat{u}_0 := \operatorname{Rec}_0(\vec{u}_0) u 0 := Rec 0 ( u 0 ) ζ J : = Π J ζ \zeta_J := \Pi_J \zeta ζ J := Π J ζ z : = ( u ⃗ 0 , c J ( ζ ) ) z := (\vec{u}_0,c_J(\zeta)) z := ( u 0 , c J ( ζ ))
By (11) and (20) ,
Γ J ( z ) = S ( u ^ 0 , ζ J ) . (37) \Gamma_J(z) = \mathcal{S}(\widehat{u}_0,\zeta_J). \tag*{(37)} Γ J ( z ) = S ( u 0 , ζ J ) . (37) Moreover, z ∈ K J , R z \in K_{J,R} z ∈ K J , R (16) . For the Cartesian grid ( z k ) k ≤ m (z^k)_{k \le m} ( z k ) k ≤ m ( ρ k ) k ≤ m (\rho_k)_{k \le m} ( ρ k ) k ≤ m I 3 I_3 I 3
A J , r ( z ) : = ∑ k = 1 m ρ k ( z ) Γ J ( z k ) . A_{J,r}(z) := \sum_{k=1}^{m} \rho_k(z)\Gamma_J(z^k). A J , r ( z ) := k = 1 ∑ m ρ k ( z ) Γ J ( z k ) . We decompose the approximation error into four terms. For any output point z ∙ ∈ D T z_\bullet\in D_T z ∙ ∈ D T
∣ S ( u 0 , ζ ) ( z ∙ ) − ∑ k = 1 m Γ J ( z k ) ( z ∙ ) ρ ~ k ( z ) ∣ ≤ ∥ S ( u 0 , ζ ) − S ( u 0 , ζ J ) ∥ Y ⏟ I 1 + ∥ S ( u 0 , ζ J ) − Γ J ( z ) ∥ Y ⏟ I 2 + ∥ Γ J ( z ) − A J , r ( z ) ∥ Y ⏟ I 3 + ∣ ∑ k = 1 m Γ J ( z k ) ( z ∙ ) ( ρ k ( z ) − ρ ~ k ( z ) ) ∣ ⏟ I 4 . (38) \begin{aligned}
\left\lvert\mathcal{S}(u_0,\zeta)(z_\bullet)-\sum_{k=1}^{m}\Gamma_J(z^k)(z_\bullet)\widetilde{\rho}_k(z)\right\rvert
&\le\underbrace{\lVert\mathcal{S}(u_0,\zeta)-\mathcal{S}(u_0,\zeta_J)\rVert_Y}_{I_1}
+\underbrace{\lVert\mathcal{S}(u_0,\zeta_J)-\Gamma_J(z)\rVert_Y}_{I_2}\\
&\quad+\underbrace{\lVert\Gamma_J(z)-A_{J,r}(z)\rVert_Y}_{I_3}
+\underbrace{\left\lvert\sum_{k=1}^{m}\Gamma_J(z^k)(z_\bullet)(\rho_k(z)-\widetilde{\rho}_k(z))\right\rvert}_{I_4}. \tag*{(38)}
\end{aligned} S ( u 0 , ζ ) ( z ∙ ) − k = 1 ∑ m Γ J ( z k ) ( z ∙ ) ρ k ( z ) ≤ I 1 ∥ S ( u 0 , ζ ) − S ( u 0 , ζ J ) ∥ Y + I 2 ∥ S ( u 0 , ζ J ) − Γ J ( z ) ∥ Y + I 3 ∥ Γ J ( z ) − A J , r ( z ) ∥ Y + I 4 k = 1 ∑ m Γ J ( z k ) ( z ∙ ) ( ρ k ( z ) − ρ k ( z )) . (38) We estimate these terms separately.
Estimates of I 1 I_1 I 1 The two solutions in I 1 I_1 I 1 (7) and the cutoff estimate (15) yield
I 1 ≤ e L F T C β 1 ∥ ( I − Π J ) ζ ∥ X 1 ≤ C 1 2 − J δ ∥ ζ ∥ X 0 ≤ C 1 2 − J δ R , I_1 \le e^{L_FT}C_{\beta_1}\lVert(I-\Pi_J)\zeta\rVert_{X_1} \le C_1 2^{-J\delta}\lVert\zeta\rVert_{X_0} \le C_1 2^{-J\delta}R, I 1 ≤ e L F T C β 1 ∥( I − Π J ) ζ ∥ X 1 ≤ C 1 2 − J δ ∥ ζ ∥ X 0 ≤ C 1 2 − J δ R , where C 1 ≥ 1 C_1 \ge1 C 1 ≥ 1 J J J R R R ζ \zeta ζ
We choose J J J
2 J δ ≥ 4 C 1 R ε B . (39) 2^{J\delta} \ge\frac{4C_1R}{\varepsilon_B}. \tag*{(39)} 2 J δ ≥ ε B 4 C 1 R . (39) Then I 1 ≤ ε B / 4 I_1 \le\varepsilon_B/4 I 1 ≤ ε B /4 4 C 1 R / ε B > 1 4C_1R/\varepsilon_B > 1 4 C 1 R / ε B > 1 J ≥ 1 J \ge1 J ≥ 1 J J J J − 1 J-1 J − 1 (39) . Thus
2 ( J − 1 ) δ < 4 C 1 R ε B , 2^{(J-1)\delta} < \frac{4C_1R}{\varepsilon_B}, 2 ( J − 1 ) δ < ε B 4 C 1 R , and consequently
2 J < 2 ( 4 C 1 R ε B ) 1 / δ , 2^J < 2\left(\frac{4C_1R}{\varepsilon_B}\right)^{1/\delta}, 2 J < 2 ( ε B 4 C 1 R ) 1/ δ , which is the first estimate in (32) . Taking the smallest J J J N J N_J N J L J L_J L J
Estimates of I 2 I_2 I 2 By (37) , the forcing in the two solutions defining I 2 I_2 I 2 (7) and (18) give
I 2 = ∥ S ( u 0 , ζ J ) − S ( u ^ 0 , ζ J ) ∥ Y ≤ e L F T ∥ u 0 − u ^ 0 ∥ ∞ ≤ C 2 L 0 δ 0 . I_2=\lVert\mathcal{S}(u_0,\zeta_J)-\mathcal{S}(\widehat{u}_0,\zeta_J)\rVert_Y \le e^{L_FT}\lVert u_0-\widehat{u}_0\rVert_\infty\le C_2L_0\delta_0. I 2 = ∥ S ( u 0 , ζ J ) − S ( u 0 , ζ J ) ∥ Y ≤ e L F T ∥ u 0 − u 0 ∥ ∞ ≤ C 2 L 0 δ 0 . Here C 2 : = e L F T C d C_2 := e^{L_FT}C_d C 2 := e L F T C d C d C_d C d (18) .
Choose
δ 0 = c 0 ε B , c 0 : = ( 4 C 2 L 0 ) − 1 . (40) \delta_0=c_0\varepsilon_B,\qquad c_0:=(4C_2L_0)^{-1}. \tag*{(40)} δ 0 = c 0 ε B , c 0 := ( 4 C 2 L 0 ) − 1 . (40) Then I 2 ≤ ε B / 4 I_2\leq\varepsilon_B/4 I 2 ≤ ε B /4
N 0 ≳ δ 0 − d ≳ ε B − d . N_0\gtrsim\delta_0^{-d}\gtrsim\varepsilon_B^{-d}. N 0 ≳ δ 0 − d ≳ ε B − d . Estimates of I 3 I_3 I 3 For z 1 = ( v ⃗ , c ) z_1=(\vec v,c) z 1 = ( v , c ) z 2 = ( v ⃗ ′ , c ′ ) z_2=(\vec v\,',c') z 2 = ( v ′ , c ′ )
∣ v ⃗ − v ⃗ ′ ∣ ℓ ∞ + ∣ c − c ′ ∣ ℓ 2 ≤ ∣ v ⃗ − v ⃗ ′ ∣ ℓ 2 + ∣ c − c ′ ∣ ℓ 2 |\vec v-\vec v\,'|_{\ell^\infty}+|c-c'|_{\ell^2}\leq|\vec v-\vec v\,'|_{\ell^2}+|c-c'|_{\ell^2} ∣ v − v ′ ∣ ℓ ∞ + ∣ c − c ′ ∣ ℓ 2 ≤ ∣ v − v ′ ∣ ℓ 2 + ∣ c − c ′ ∣ ℓ 2 ≤ 2 ∣ z 1 − z 2 ∣ ℓ 2 . \leq2|z_1-z_2|_{\ell^2}. ≤ 2∣ z 1 − z 2 ∣ ℓ 2 . Thus Lemma 3.5 shows that Γ J \Gamma_J Γ J 2 L J 2L_J 2 L J K J , R K_{J,R} K J , R r / 2 r/2 r /2 N = N 0 + N J N=N_0+N_J N = N 0 + N J a ∈ { M , R J } a\in\{M,R_J\} a ∈ { M , R J }
n a : = ⌈ 2 a N r ⌉ , G a : = { − a + ( 2 j − 1 ) a n a : j = 1 , … , n a } . n_a:=\left\lceil\frac{2a\sqrt{N}}{r}\right\rceil,\qquad
G_a:=\left\{-a+\frac{(2j-1)a}{n_a}:j=1,\ldots,n_a\right\}. n a := ⌈ r 2 a N ⌉ , G a := { − a + n a ( 2 j − 1 ) a : j = 1 , … , n a } . Let
{ z 1 , … , z m } : = G M N 0 × G R J N J ⊂ K J , R . \{z^1,\ldots,z^m\}:=G_M^{N_0}\times G_{R_J}^{N_J}\subset K_{J,R}. { z 1 , … , z m } := G M N 0 × G R J N J ⊂ K J , R . For every z ∈ K J , R z\in K_{J,R} z ∈ K J , R z k z^k z k
∣ z − z k ∣ ℓ ∞ ≤ r 2 N . |z-z^k|_{\ell^\infty}\leq\frac{r}{2\sqrt{N}}. ∣ z − z k ∣ ℓ ∞ ≤ 2 N r . Consequently,
∣ z − z k ∣ ℓ 2 ≤ N ∣ z − z k ∣ ℓ ∞ ≤ r 2 , |z-z^k|_{\ell^2}\leq\sqrt{N}|z-z^k|_{\ell^\infty}\leq\frac{r}{2}, ∣ z − z k ∣ ℓ 2 ≤ N ∣ z − z k ∣ ℓ ∞ ≤ 2 r , so these centers form an r / 2 r/2 r /2 K J , R K_{J,R} K J , R
m = n M N 0 n R J N J ≤ ( 1 + 2 ( M ∨ R J ) N r ) N , m=n_M^{N_0}n_{R_J}^{N_J}\leq\left(1+\frac{2(M\vee R_J)\sqrt{N}}{r}\right)^N, m = n M N 0 n R J N J ≤ ( 1 + r 2 ( M ∨ R J ) N ) N , which proves (35) . We now define interpolation weights on this grid. Write G a = { ξ a , j : 1 ≤ j ≤ n a } G_a=\{\xi_{a,j}:1\leq j\leq n_a\} G a = { ξ a , j : 1 ≤ j ≤ n a } s a : = 2 a / n a ≤ r / N s_a:=2a/n_a\leq r/\sqrt{N} s a := 2 a / n a ≤ r / N n a = 1 n_a=1 n a = 1 b a , 1 ≡ 1 b_{a,1}\equiv1 b a , 1 ≡ 1 n a ≥ 2 n_a\geq2 n a ≥ 2
b a , 1 ( s ) : = P ( ξ a , 2 − s s a ) , b_{a,1}(s):=P\left(\frac{\xi_{a,2}-s}{s_a}\right), b a , 1 ( s ) := P ( s a ξ a , 2 − s ) , b a , n a ( s ) : = P ( s − ξ a , n a − 1 s a ) , b_{a,n_a}(s):=P\left(\frac{s-\xi_{a,n_a-1}}{s_a}\right), b a , n a ( s ) := P ( s a s − ξ a , n a − 1 ) , b a , j ( s ) : = ( 1 − ∣ s − ξ a , j ∣ s a ) + , 1 < j < n a , b_{a,j}(s):=\left(1-\frac{|s-\xi_{a,j}|}{s_a}\right)_+,\qquad1<j<n_a, b a , j ( s ) := ( 1 − s a ∣ s − ξ a , j ∣ ) + , 1 < j < n a , where
P ( v ) : = { 0 , v < 0 , v , 0 ≤ v ≤ 1 , 1 , v > 1. P(v):=
\begin{cases}
0, & v<0,\\
v, & 0\leq v\leq1,\\
1, & v>1.
\end{cases} P ( v ) := ⎩ ⎨ ⎧ 0 , v , 1 , v < 0 , 0 ≤ v ≤ 1 , v > 1. These functions interpolate the grid values and are extended constantly between the first or last grid point and the corresponding boundary. For s ∈ [ − a , a ] s\in[-a,a] s ∈ [ − a , a ]
b a , j ( s ) ≥ 0 , ∑ j = 1 n a b a , j ( s ) = 1 , b a , j ( s ) > 0 ⟹ ∣ s − ξ a , j ∣ ≤ s a . b_{a,j}(s)\geq0,\qquad\sum_{j=1}^{n_a}b_{a,j}(s)=1,\qquad b_{a,j}(s)>0\Longrightarrow|s-\xi_{a,j}|\leq s_a. b a , j ( s ) ≥ 0 , j = 1 ∑ n a b a , j ( s ) = 1 , b a , j ( s ) > 0 ⟹ ∣ s − ξ a , j ∣ ≤ s a . The last bound also holds when n a = 1 n_a=1 n a = 1 ξ a , 1 = 0 \xi_{a,1}=0 ξ a , 1 = 0 ∣ s ∣ ≤ a ≤ s a |s|\leq a\leq s_a ∣ s ∣ ≤ a ≤ s a
Set a q = M a_q=M a q = M 1 ≤ q ≤ N 0 1\le q\le N_0 1 ≤ q ≤ N 0 a q = R J a_q=R_J a q = R J N 0 < q ≤ N N_0<q\le N N 0 < q ≤ N α = ( j 1 , … , j N ) \alpha=(j_1,\ldots,j_N) α = ( j 1 , … , j N )
z α : = ( ξ a q , j q ) q = 1 N , ρ α ( w ) : = ∏ q = 1 N b a q , j q ( w q ) . (41) z^\alpha:=(\xi_{a_q,j_q})_{q=1}^N,\qquad\rho_\alpha(w):=\prod_{q=1}^N b_{a_q,j_q}(w_q). \tag*{(41)} z α := ( ξ a q , j q ) q = 1 N , ρ α ( w ) := q = 1 ∏ N b a q , j q ( w q ) . (41) For each center z k = z α z^k=z^\alpha z k = z α ρ k : = ρ α \rho_k:=\rho_\alpha ρ k := ρ α K J , R K_{J,R} K J , R w ∈ K J , R w\in K_{J,R} w ∈ K J , R
ρ k ( w ) > 0 ⟹ ∣ w − z k ∣ 2 ≤ ( ∑ q = 1 N s a q 2 ) 1 / 2 ≤ r . \rho_k(w)>0\quad\Longrightarrow\quad\lvert w-z^k\rvert_2\le\left(\sum_{q=1}^N s_{a_q}^2\right)^{1/2}\le r. ρ k ( w ) > 0 ⟹ ∣ w − z k ∣ 2 ≤ ( q = 1 ∑ N s a q 2 ) 1/2 ≤ r . The 2 L J 2L_J 2 L J Γ J \Gamma_J Γ J
I 3 ≤ ∑ k = 1 m ρ k ( z ) ∥ Γ J ( z ) − Γ J ( z k ) ∥ Y ≤ 2 L J r . I_3\le\sum_{k=1}^m\rho_k(z)\lVert\Gamma_J(z)-\Gamma_J(z^k)\rVert_Y\le2L_Jr. I 3 ≤ k = 1 ∑ m ρ k ( z ) ∥ Γ J ( z ) − Γ J ( z k ) ∥ Y ≤ 2 L J r . Choosing
r : = ε B 8 L J (42) r:=\frac{\varepsilon_B}{8L_J} \tag*{(42)} r := 8 L J ε B (42) therefore gives I 3 ≤ ε B / 4 I_3\le\varepsilon_B/4 I 3 ≤ ε B /4
Estimates of I 4 I_4 I 4 Write z k = ( v ⃗ k , c k ) z^k=(\vec v^k,c^k) z k = ( v k , c k ) z k ∈ K J , R z^k\in K_{J,R} z k ∈ K J , R v i k v_i^k v i k [ − M , M ] [-M,M] [ − M , M ] Rec 0 ( v ⃗ k ) \operatorname{Rec}_0(\vec v^k) Rec 0 ( v k )
∥ Rec 0 ( v ⃗ k ) ∥ ∞ ≤ M . \lVert\operatorname{Rec}_0(\vec v^k)\rVert_\infty\le M. ∥ Rec 0 ( v k ) ∥ ∞ ≤ M . Similarly, ∣ c k ∣ 2 ≤ N J R J \lvert c^k\rvert_2\le\sqrt{N_J}R_J ∣ c k ∣ 2 ≤ N J R J
∥ Rec J ( c k ) ∥ X 1 ≤ C N J 1 / 2 ∣ c k ∣ 2 ≤ C N J R J . \lVert\operatorname{Rec}_J(c^k)\rVert_{X_1}\le C N_J^{1/2}\lvert c^k\rvert_2\le C N_JR_J. ∥ Rec J ( c k ) ∥ X 1 ≤ C N J 1/2 ∣ c k ∣ 2 ≤ C N J R J . The a priori estimate (7) now implies, for a constant C B ≥ 1 C_B\ge1 C B ≥ 1
max k ≤ m ∥ Γ J ( z k ) ∥ Y ≤ B J : = C B ( 1 + N J R J ) . (43) \max_{k\le m}\lVert\Gamma_J(z^k)\rVert_Y\le B_J:=C_B(1+N_JR_J). \tag*{(43)} k ≤ m max ∥ Γ J ( z k ) ∥ Y ≤ B J := C B ( 1 + N J R J ) . (43) Each one-dimensional function b a , j b_{a,j} b a , j n a = 1 n_a=1 n a = 1 [ 0 , 1 ] [0,1] [ 0 , 1 ] N N N δ B ∈ ( 0 , 1 ) \delta_B\in(0,1) δ B ∈ ( 0 , 1 ) ρ ^ k \widehat{\rho}_k ρ k [ 0 , 1 ] [0,1] [ 0 , 1 ]
max k ≤ m ∥ ρ k − ρ ^ k ∥ C ( K J , R ) ≤ δ B . (44) \max_{k\le m}\lVert\rho_k-\widehat{\rho}_k\rVert_{C(K_{J,R})}\le\delta_B. \tag*{(44)} k ≤ m max ∥ ρ k − ρ k ∥ C ( K J , R ) ≤ δ B . (44) For N = 1 N=1 N = 1 ρ ^ k \widehat{\rho}_k ρ k
depth ( ρ ^ k ) ≤ C log ( e N ) log e N δ B , size ( ρ ^ k ) ≤ C N log e N δ B , (45) \begin{aligned}
\operatorname{depth}(\widehat{\rho}_k)&\le C\log(eN)\log\frac{eN}{\delta_B},\\
\operatorname{size}(\widehat{\rho}_k)&\le CN\log\frac{eN}{\delta_B}, \tag*{(45)}
\end{aligned} depth ( ρ k ) size ( ρ k ) ≤ C log ( e N ) log δ B e N , ≤ C N log δ B e N , (45) where size denotes the total number of nonzero weights and biases. The m m m [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ]
Let P K J , R P_{K_J,R} P K J , R K J , R K_{J,R} K J , R K J , R K_{J,R} K J , R K J , R K_{J,R} K J , R
ρ ~ k : = ρ ^ k ∘ P K J , R . (46) \widetilde{\rho}_k := \widehat{\rho}_k \circ P_{K_J,R}. \tag*{(46)} ρ k := ρ k ∘ P K J , R . (46) Lemma 3.9(1) shows that this is again a ReLU network. Since P K J , R P_{K_J,R} P K J , R K J , R K_{J,R} K J , R (44) gives, for the last term in (38) ,
I 4 ≤ ∑ k = 1 m ∥ Γ J ( z k ) ∥ Y ∣ ρ k ( z ) − ρ ~ k ( z ) ∣ ≤ m B J δ B . \begin{aligned}
I_4 &\le\sum_{k=1}^{m} \|\Gamma_J(z^k)\|_Y \lvert\rho_k(z)-\widetilde{\rho}_k(z)\rvert\\
&\le mB_J\delta_B.
\end{aligned} I 4 ≤ k = 1 ∑ m ∥ Γ J ( z k ) ∥ Y ∣ ρ k ( z ) − ρ k ( z )∣ ≤ m B J δ B . Set
δ B : = ε B 4 m ( B J + 1 ) . (47) \delta_B := \frac{\varepsilon_B}{4m(B_J+1)}. \tag*{(47)} δ B := 4 m ( B J + 1 ) ε B . (47) Then I 4 ≤ ε B / 4 I_4 \le\varepsilon_B/4 I 4 ≤ ε B /4 w ∈ R N 0 + N J w \in\mathbb{R}^{N_0+N_J} w ∈ R N 0 + N J
∑ k = 1 m ∣ ρ ~ k ( w ) ∣ ≤ ∑ k = 1 m ρ k ( P K J , R w ) + m δ B ≤ 1 + ε B 4 ( B J + 1 ) ≤ 2 , \begin{aligned}
\sum_{k=1}^{m} \lvert\widetilde{\rho}_k(w)\rvert
&\le\sum_{k=1}^{m}\rho_k(P_{K_J,R}w)+m\delta_B \\
&\le1+\frac{\varepsilon_B}{4(B_J+1)}\le2,
\end{aligned} k = 1 ∑ m ∣ ρ k ( w )∣ ≤ k = 1 ∑ m ρ k ( P K J , R w ) + m δ B ≤ 1 + 4 ( B J + 1 ) ε B ≤ 2 , which proves (34) . For the choice (47) ,
log e N δ B = log 4 e N m ( B J + 1 ) ε B ≤ C H B . \log\frac{e^N}{\delta_B}
= \log\frac{4eNm(B_J+1)}{\varepsilon_B}
\le CH_B. log δ B e N = log ε B 4 e N m ( B J + 1 ) ≤ C H B . The coordinate projection requires O ( N ) O(N) O ( N ) (45) gives the stated bounds for each ρ ~ k \widetilde{\rho}_k ρ k (36) . Since their outputs are nonnegative, their final affine outputs may be treated as ReLU hidden activations. Appending an identity output map adds at most m m m
Combining the estimates from (39) , (40) , (42) , and (47) , give I i ≤ ε B / 4 I_i \le\varepsilon_B/4 I i ≤ ε B /4 1 ≤ i ≤ 4 1 \le i \le4 1 ≤ i ≤ 4 (38) proves (33) , uniformly in u 0 u_0 u 0 ζ \zeta ζ z ∙ z_\bullet z ∙
Operator approximation
We now combine the function approximation of Theorem 4.1 with the functional approximation of Theorem 4.2 . This is the SPDE counterpart of the final operator approximation step in [5 ] .
Theorem 4.3. Let d ≥ 2 d \ge2 d ≥ 2 F F F β 1 \beta_1 β 1 (2) and γ \gamma γ (9) . For every ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) J J J N 0 N_0 N 0
sup u 0 ∈ G 0 E [ sup ( t , x ) ∈ D T ∣ S ( u 0 , η ) ( t , x ) − ∑ ℓ = 1 N 1 a ~ ℓ ( u ⃗ 0 , c J ( η ) ) q ~ ℓ ( t , x ) ∣ ] ≤ ε . (48) \sup_{u_0\in\mathcal{G}_0}\mathbb{E}\left[\sup_{(t,x)\in D_T}\left|S(u_0,\eta)(t,x)-\sum_{\ell=1}^{N_1}\widetilde{a}_\ell(\vec{u}_0,c_J(\eta))\widetilde{q}_\ell(t,x)\right|\right]\le\varepsilon. \tag*{(48)} u 0 ∈ G 0 sup E [ ( t , x ) ∈ D T sup S ( u 0 , η ) ( t , x ) − ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( η )) q ℓ ( t , x ) ] ≤ ε . (48) The parameters may be chosen so that, with R = C 0 log ( e / ε ) R=C_0\sqrt{\log(e/\varepsilon)} R = C 0 log ( e / ε )
N J ≤ C ( R ε ) ( d + 2 ) / δ , (49) N_J \le C\left(\frac{R}{\varepsilon}\right)^{(d+2)/\delta}, \tag*{(49)} N J ≤ C ( ε R ) ( d + 2 ) / δ , (49) N 0 ≤ C ε − d , (50) N_0 \le C\varepsilon^{-d}, \tag*{(50)} N 0 ≤ C ε − d , (50) N 1 ≤ C ( 1 + R ε ) ( d + 2 ) / γ . (51) N_1 \le C\left(\frac{1+R}{\varepsilon}\right)^{(d+2)/\gamma}. \tag*{(51)} N 1 ≤ C ( ε 1 + R ) ( d + 2 ) / γ . (51) For each trunk network, its depth L T , ℓ L_{T,\ell} L T , ℓ S T , ℓ S_{T,\ell} S T , ℓ
L T , ℓ ≤ C log ( e / ε ) , S T , ℓ ≤ C log ( e / ε ) , 1 ≤ ℓ ≤ N 1 . (52) L_{T,\ell} \le C\log(e/\varepsilon), \qquad S_{T,\ell} \le C\log(e/\varepsilon), \qquad1 \le\ell\le N_1. \tag*{(52)} L T , ℓ ≤ C log ( e / ε ) , S T , ℓ ≤ C log ( e / ε ) , 1 ≤ ℓ ≤ N 1 . (52) The branch networks can be realized as the coordinates of a single network a ~ : R N → R N 1 \tilde{a}:\mathbb{R}^{N}\to\mathbb{R}^{N_1} a ~ : R N → R N 1 N = N 0 + N J N=N_0+N_J N = N 0 + N J m m m
m ≤ ( 1 + 2 ( M ∨ R J ) N r ) N , r = ε 48 L J . (53) m \le\left(1+\frac{2(M\vee R_J)\sqrt{N}}{r}\right)^N, \qquad r=\frac{\varepsilon}{48L_J}. \tag*{(53)} m ≤ ( 1 + r 2 ( M ∨ R J ) N ) N , r = 48 L J ε . (53) Set B J : = C B ( 1 + N J R J ) B_J:=C_B(1+N_JR_J) B J := C B ( 1 + N J R J ) C B C_B C B Theorem 4.2 , and define
H ε : = log ( e N m ( B J + 1 ) ε ) . H_\varepsilon:=\log\left(\frac{eNm(B_J+1)}{\varepsilon}\right). H ε := log ( ε e N m ( B J + 1 ) ) . The shared branch network has depth L B L_B L B S B S_B S B
L B ≤ C log ( e N ) H ε , S B ≤ C ( m N H ε + m N 1 ) . (54) L_B\le C\log(eN)H_\varepsilon,\qquad S_B\le C(mNH_\varepsilon+mN_1). \tag*{(54)} L B ≤ C log ( e N ) H ε , S B ≤ C ( m N H ε + m N 1 ) . (54) Here the shared partition weights are counted once, and m N 1 mN_1 m N 1
S B + ∑ ℓ = 1 N 1 S T , ℓ ≤ C ( m N H ε + m N 1 + N 1 log ( e / ε ) ) . (55) S_B+\sum_{\ell=1}^{N_1}S_{T,\ell}\le C(mNH_\varepsilon+mN_1+N_1\log(e/\varepsilon)). \tag*{(55)} S B + ℓ = 1 ∑ N 1 S T , ℓ ≤ C ( m N H ε + m N 1 + N 1 log ( e / ε )) . (55) The constants in these size bounds are independent of ε \varepsilon ε
Proof. Construction and uniform approximation. For a radius R ≥ 1 R\ge1 R ≥ 1
Ω R : = { ∥ η ∥ X 0 ≤ R } . \Omega_R:=\{\|\eta\|_{X_0}\le R\}. Ω R := { ∥ η ∥ X 0 ≤ R } . We first construct an approximation whose error on Ω R \Omega_R Ω R ε / 2 \varepsilon/2 ε /2 ε T \varepsilon_T ε T ε B \varepsilon_B ε B ℓ 1 \ell^1 ℓ 1 (29) , it is enough to require
ε T + 3 2 ε B ≤ ε 2 . (56) \varepsilon_T+\frac{3}{2}\varepsilon_B\le\frac{\varepsilon}{2}. \tag*{(56)} ε T + 2 3 ε B ≤ 2 ε . (56) We use the convenient choice
ε T : = ε 4 , ε B : = ε 6 , (57) \varepsilon_T:=\frac{\varepsilon}{4},\qquad\varepsilon_B:=\frac{\varepsilon}{6}, \tag*{(57)} ε T := 4 ε , ε B := 6 ε , (57) for which equality holds in (56) .
Consider the class of output functions generated on the noise ball,
V R : = { S ( u 0 , ζ ) : u 0 ∈ G 0 , ζ ∈ X 0 , ∥ ζ ∥ X 0 ≤ R } . \mathcal{V}_R:=\{S(u_0,\zeta):u_0\in G_0,\ \zeta\in X_0,\ \|\zeta\|_{X_0}\le R\}. V R := { S ( u 0 , ζ ) : u 0 ∈ G 0 , ζ ∈ X 0 , ∥ ζ ∥ X 0 ≤ R } . Lemma 2.7 gives, for every v ∈ V R v\in\mathcal{V}_R v ∈ V R
∥ v ∥ ∞ ≤ B ( R ) , ∣ v ( y ) − v ( y ′ ) ∣ ≤ Λ ( R ) d s ( y , y ′ ) γ . \|v\|_\infty\le B(R),\qquad|v(y)-v(y')|\le\Lambda(R)d_s(y,y')^\gamma. ∥ v ∥ ∞ ≤ B ( R ) , ∣ v ( y ) − v ( y ′ ) ∣ ≤ Λ ( R ) d s ( y , y ′ ) γ . Apply Theorem 4.1 to V R \mathcal{V}_R V R ε T \varepsilon_T ε T (57) ; the hypothesis ε T ≤ 2 Λ ( R ) \varepsilon_T \le2\Lambda(R) ε T ≤ 2Λ ( R ) ε T < 1 ≤ Λ ( R ) \varepsilon_T < 1 \le\Lambda(R) ε T < 1 ≤ Λ ( R ) ε T ≤ B ( R ) \varepsilon_T \le B(R) ε T ≤ B ( R ) B ( R ) ≥ 1 B(R) \ge1 B ( R ) ≥ 1 y 1 , … , y N 1 y_1,\ldots,y_{N_1} y 1 , … , y N 1 q ~ 1 , … , q ~ N 1 \widetilde{q}_1,\ldots,\widetilde{q}_{N_1} q 1 , … , q N 1
sup v ∈ V R ∥ v − ∑ ℓ = 1 N 1 v ( y ℓ ) q ~ ℓ ∥ Y ≤ ε T , (58) \sup_{v\in\mathcal{V}_R}\left\|v-\sum_{\ell=1}^{N_1}v(y_\ell)\widetilde{q}_\ell\right\|_Y\le\varepsilon_T, \tag*{(58)} v ∈ V R sup v − ℓ = 1 ∑ N 1 v ( y ℓ ) q ℓ Y ≤ ε T , (58) sup y ∈ D T ∑ ℓ = 1 N 1 ∣ q ~ ℓ ( y ) ∣ ≤ 3 2 . (59) \sup_{y\in D_T}\sum_{\ell=1}^{N_1}\left|\widetilde{q}_\ell(y)\right|\le\frac{3}{2}. \tag*{(59)} y ∈ D T sup ℓ = 1 ∑ N 1 ∣ q ℓ ( y ) ∣ ≤ 2 3 . (59) The nodes and the trunk networks depend on R R R ε \varepsilon ε u 0 u_0 u 0
Next, apply Theorem 4.2 with the same radius R R R ε B \varepsilon_B ε B (57) . This determines J J J δ 0 \delta_0 δ 0 K J , R K_{J,R} K J , R z 1 , … , z m z^1,\ldots,z^m z 1 , … , z m ρ ~ 1 , … , ρ ~ m \widetilde{\rho}_1,\ldots,\widetilde{\rho}_m ρ 1 , … , ρ m y ∈ D T y\in D_T y ∈ D T
sup u 0 ∈ G 0 sup ∥ ζ ∥ X 0 ≤ R ∣ S ( u 0 , ζ ) ( y ) − ∑ k = 1 m Γ J ( z k ) ( y ) ρ ~ k ( u ⃗ 0 , c J ( ζ ) ) ∣ ≤ ε B . (60) \sup_{u_0\in\mathcal{G}_0}\sup_{\|\zeta\|_{X_0}\le R}\left|\mathcal{S}(u_0,\zeta)(y)-\sum_{k=1}^{m}\Gamma_J(z^k)(y)\widetilde{\rho}_k(\vec{u}_0,c_J(\zeta))\right|\le\varepsilon_B. \tag*{(60)} u 0 ∈ G 0 sup ∥ ζ ∥ X 0 ≤ R sup S ( u 0 , ζ ) ( y ) − k = 1 ∑ m Γ J ( z k ) ( y ) ρ k ( u 0 , c J ( ζ )) ≤ ε B . (60) For each trunk node y ℓ y_\ell y ℓ
a ~ ℓ ( w ) : = ∑ k = 1 m Γ J ( z k ) ( y ℓ ) ρ ~ k ( w ) , w ∈ R N 0 + N J . (61) \widetilde{a}_\ell(w):=\sum_{k=1}^{m}\Gamma_J(z^k)(y_\ell)\widetilde{\rho}_k(w),\qquad w\in\mathbb{R}^{N_0+N_J}. \tag*{(61)} a ℓ ( w ) := k = 1 ∑ m Γ J ( z k ) ( y ℓ ) ρ k ( w ) , w ∈ R N 0 + N J . (61) These branch networks can be combined into a single vector-valued network:
a ~ ( w ) = A ρ ~ ( w ) , A ℓ k : = Γ J ( z k ) ( y ℓ ) , A ∈ R N 1 × m . \widetilde{a}(w)=A\widetilde{\rho}(w),\qquad A_{\ell k}:=\Gamma_J(z^k)(y_\ell),\qquad A\in\mathbb{R}^{N_1\times m}. a ( w ) = A ρ ( w ) , A ℓ k := Γ J ( z k ) ( y ℓ ) , A ∈ R N 1 × m . Theorem 4.2 realizes the shared weights at a common depth with nonnegative outputs as the final hidden activations. Appending the linear map A A A m N 1 mN_1 m N 1 (60) at y = y ℓ y=y_\ell y = y ℓ
max ℓ ≤ N 1 ∣ S ( u 0 , ζ ) ( y ℓ ) − a ~ ℓ ( u ⃗ 0 , c J ( ζ ) ) ∣ ≤ ε B (62) \max_{\ell\le N_1}\left|\mathcal{S}(u_0,\zeta)(y_\ell)-\widetilde{a}_\ell(\vec{u}_0,c_J(\zeta))\right|\le\varepsilon_B \tag*{(62)} ℓ ≤ N 1 max ∣ S ( u 0 , ζ ) ( y ℓ ) − a ℓ ( u 0 , c J ( ζ )) ∣ ≤ ε B (62) whenever u 0 ∈ G 0 u_0\in\mathcal{G}_0 u 0 ∈ G 0 ∥ ζ ∥ X 0 ≤ R \|\zeta\|_{X_0}\le R ∥ ζ ∥ X 0 ≤ R
We now combine these approximations. Fix u 0 ∈ G 0 u_0\in\mathcal{G}_0 u 0 ∈ G 0 Ω R \Omega_R Ω R u : = S ( u 0 , η ) u:=\mathcal{S}(u_0,\eta) u := S ( u 0 , η ) y ∈ D T y\in D_T y ∈ D T
∣ u ( y ) − ∑ ℓ = 1 N 1 a ~ ℓ ( u ⃗ 0 , c J ( η ) ) q ~ ℓ ( y ) ∣ ≤ ∣ u ( y ) − ∑ ℓ = 1 N 1 u ( y ℓ ) q ~ ℓ ( y ) ∣ + ∑ ℓ = 1 N 1 ∣ u ( y ℓ ) − a ~ ℓ ( u ⃗ 0 , c J ( η ) ) ∣ ∣ q ~ ℓ ( y ) ∣ . \left|u(y)-\sum_{\ell=1}^{N_1}\widetilde{a}_\ell(\vec{u}_0,c_J(\eta))\widetilde{q}_\ell(y)\right|
\le
\left|u(y)-\sum_{\ell=1}^{N_1}u(y_\ell)\widetilde{q}_\ell(y)\right|
+\sum_{\ell=1}^{N_1}\left|u(y_\ell)-\widetilde{a}_\ell(\vec{u}_0,c_J(\eta))\right|\left|\widetilde{q}_\ell(y)\right|. u ( y ) − ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( η )) q ℓ ( y ) ≤ u ( y ) − ℓ = 1 ∑ N 1 u ( y ℓ ) q ℓ ( y ) + ℓ = 1 ∑ N 1 ∣ u ( y ℓ ) − a ℓ ( u 0 , c J ( η )) ∣ ∣ q ℓ ( y ) ∣ . The first term is bounded by (58) ; the second is bounded using (62) and (59) . Taking the supremum over y y y
∥ u − ∑ ℓ = 1 N 1 a ~ ℓ ( u ⃗ 0 , c J ( η ) ) q ~ ℓ ∥ Y ≤ ε T + 3 2 ε B ≤ ε 2 . (63) \left\|u-\sum_{\ell=1}^{N_1}\widetilde{a}_\ell(\vec{u}_0,c_J(\eta))\widetilde{q}_\ell\right\|_Y\le\varepsilon_T+\frac{3}{2}\varepsilon_B\le\frac{\varepsilon}{2}. \tag*{(63)} u − ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( η )) q ℓ Y ≤ ε T + 2 3 ε B ≤ 2 ε . (63) This is the deterministic branch–trunk approximation on Ω R \Omega_R Ω R
Expected approximation error. The coordinatewise projection in (45) ensures that the branch network remains bounded even when the encoded forcing lies outside K J , R K_{J,R} K J , R w ∈ R N 0 + N J w \in\mathbb{R}^{N_0+N_J} w ∈ R N 0 + N J
∣ a ~ ℓ ( w ) ∣ ≤ B J ∑ k = 1 m ∣ ρ ~ k ( w ) ∣ ≤ 2 B J . |\widetilde{a}_{\ell}(w)| \le B_J \sum_{k=1}^{m} |\widetilde{\rho}_k(w)| \le2B_J. ∣ a ℓ ( w ) ∣ ≤ B J k = 1 ∑ m ∣ ρ k ( w ) ∣ ≤ 2 B J . Combining this with (58),
sup y ∈ D T ∣ ∑ ℓ = 1 N 1 a ~ ℓ ( w ) q ~ ℓ ( y ) ∣ ≤ 3 B J . (64) \sup_{y\in D_T} \left|\sum_{\ell=1}^{N_1} \widetilde{a}_{\ell}(w)\widetilde{q}_{\ell}(y)\right| \le3B_J. \tag*{(64)} y ∈ D T sup ℓ = 1 ∑ N 1 a ℓ ( w ) q ℓ ( y ) ≤ 3 B J . (64) We next express B J B_J B J R R R ε \varepsilon ε R J R_J R J
B J ≤ C ( 1 + N J 2 − J β 0 R ) ≤ C ( 1 + R 2 J ( d + 2 − β 0 ) ) . B_J \le C(1+N_J2^{-J\beta_0}R)
\le C(1+R2^{J(d+2-\beta_0)}). B J ≤ C ( 1 + N J 2 − J β 0 R ) ≤ C ( 1 + R 2 J ( d + 2 − β 0 ) ) . Since β 0 > − 2 \beta_0>-2 β 0 > − 2 d + 2 − β 0 < d + 4 d+2-\beta_0<d+4 d + 2 − β 0 < d + 4 J J J ε B = ε / 6 \varepsilon_B=\varepsilon/6 ε B = ε /6
B J ≤ C [ 1 + R ( R ε ) ( d + 4 ) / δ ] . (65) B_J \le C\left[1+R\left(\frac{R}{\varepsilon}\right)^{(d+4)/\delta}\right]. \tag*{(65)} B J ≤ C [ 1 + R ( ε R ) ( d + 4 ) / δ ] . (65) Let
E θ ( u 0 , η ) : = ∥ S ( u 0 , η ) − ∑ ℓ = 1 N 1 a ~ ℓ ( u ~ 0 , c J ( η ) ) q ~ ℓ ∥ Y . \mathcal{E}_{\theta}(u_0,\eta):=
\left\|S(u_0,\eta)-\sum_{\ell=1}^{N_1}\widetilde{a}_{\ell}(\widetilde{u}_0,c_J(\eta))\widetilde{q}_{\ell}\right\|_Y. E θ ( u 0 , η ) := S ( u 0 , η ) − ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( η )) q ℓ Y . For each fixed u 0 ∈ G 0 u_0\in\mathcal{G}_0 u 0 ∈ G 0 η ↦ E θ ( u 0 , η ) \eta\mapsto\mathcal{E}_{\theta}(u_0,\eta) η ↦ E θ ( u 0 , η ) X 0 X_0 X 0 S S S C ( T d ) × X 1 C(\mathbb{T}^d)\times X_1 C ( T d ) × X 1 Y Y Y X 0 ↪ X 1 X_0\hookrightarrow X_1 X 0 ↪ X 1 Y Y Y
E θ ( u 0 , η ) ≤ C ( 1 + B J + ∥ η ∥ X 0 ) , (66) \mathcal{E}_{\theta}(u_0,\eta)\le C(1+B_J+\|\eta\|_{X_0}), \tag*{(66)} E θ ( u 0 , η ) ≤ C ( 1 + B J + ∥ η ∥ X 0 ) , (66) uniformly in u 0 ∈ G 0 u_0\in\mathcal{G}_0 u 0 ∈ G 0 p : = ( d + 4 ) / δ + 1 p:=(d+4)/\delta+1 p := ( d + 4 ) / δ + 1 C C C
1 + B J ≤ C ε − p ( 1 + R p ) . 1+B_J\le C\varepsilon^{-p}(1+R^p). 1 + B J ≤ C ε − p ( 1 + R p ) . Lemmas 2.3 and 2.3, applied to (65), now give constants c 1 , C > 0 c_1,C>0 c 1 , C > 0
sup u 0 ∈ G 0 E [ E θ ( u 0 , η ) 1 Ω R c ] ≤ C ε − p ( 1 + R p ) e − c 1 R 2 . (67) \sup_{u_0\in\mathcal{G}_0}\mathbb{E}[\mathcal{E}_{\theta}(u_0,\eta)1_{\Omega_R^c}]
\le C\varepsilon^{-p}(1+R^p)e^{-c_1R^2}. \tag*{(67)} u 0 ∈ G 0 sup E [ E θ ( u 0 , η ) 1 Ω R c ] ≤ C ε − p ( 1 + R p ) e − c 1 R 2 . (67) The constants C , c 1 C,c_1 C , c 1 R ≥ 1 R\ge1 R ≥ 1
R : = C 0 log ( e / ε ) . R:=C_0\sqrt{\log(e/\varepsilon)}. R := C 0 log ( e / ε ) . The right-hand side of (66) is bounded by
C ε − p ( 1 + C 0 p log ( e / ε ) p / 2 ) ( e / ε ) − c 1 C 0 2 . C\varepsilon^{-p}(1+C_0^p\log(e/\varepsilon)^{p/2})(e/\varepsilon)^{-c_1C_0^2}. C ε − p ( 1 + C 0 p log ( e / ε ) p /2 ) ( e / ε ) − c 1 C 0 2 . Taking C 0 C_0 C 0 c 1 C 0 2 > p + 2 c_1C_0^2>p+2 c 1 C 0 2 > p + 2 ε / 2 \varepsilon/2 ε /2 ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 )
sup u 0 ∈ G 0 E [ E θ ( u 0 , η ) ] ≤ ε 2 P ( Ω R ) + sup u 0 ∈ G 0 E [ E θ ( u 0 , η ) 1 Ω R c ] ≤ ε , \sup_{u_0\in\mathcal{G}_0}\mathbb{E}[\mathcal{E}_{\theta}(u_0,\eta)]
\le\frac{\varepsilon}{2}\mathbb{P}(\Omega_R)
+\sup_{u_0\in\mathcal{G}_0}\mathbb{E}[\mathcal{E}_{\theta}(u_0,\eta)1_{\Omega_R^c}]
\le\varepsilon, u 0 ∈ G 0 sup E [ E θ ( u 0 , η )] ≤ 2 ε P ( Ω R ) + u 0 ∈ G 0 sup E [ E θ ( u 0 , η ) 1 Ω R c ] ≤ ε , which proves (48) .
Measurement counts and network sizes. With ε B = ε / 6 \varepsilon_B = \varepsilon/6 ε B = ε /6 (39) gives
2 J ≤ C ( R ε ) 1 / δ . 2^J \le C\left(\frac{R}{\varepsilon}\right)^{1/\delta}. 2 J ≤ C ( ε R ) 1/ δ . Using N J ≍ 2 J ( d + 2 ) N_J \asymp2^{J(d+2)} N J ≍ 2 J ( d + 2 ) (49) . Likewise, (40) gives δ 0 ≍ ε \delta_0 \asymp\varepsilon δ 0 ≍ ε N 0 ≍ δ 0 − d N_0 \asymp\delta_0^{-d} N 0 ≍ δ 0 − d (50) . Finally, Lemma 2.7 gives Λ ( R ) ≤ C ( 1 + R ) \Lambda(R) \le C(1+R) Λ ( R ) ≤ C ( 1 + R ) (30) with ε T = ε / 4 \varepsilon_T = \varepsilon/4 ε T = ε /4 (51) .
For each trunk network, Theorem 4.1 bounds its depth and number of nonzero parameters by
C log ( 4 B ( R ) N 1 ε ) . C\log\left(\frac{4B(R)N_1}{\varepsilon}\right). C log ( ε 4 B ( R ) N 1 ) . Since B ( R ) ≤ C ( 1 + R ) B(R) \le C(1+R) B ( R ) ≤ C ( 1 + R ) (51) implies that this is at most C log ( e ( 1 + R ) / ε ) C\log(e(1+R)/\varepsilon) C log ( e ( 1 + R ) / ε ) d d d γ \gamma γ R = C 0 log ( e / ε ) R=C_0\sqrt{\log(e/\varepsilon)} R = C 0 log ( e / ε ) (52) .
For the shared branch network, the choice ε B = ε / 6 \varepsilon_B=\varepsilon/6 ε B = ε /6 r = ε / ( 48 L J ) r=\varepsilon/(48L_J) r = ε / ( 48 L J ) (53) . It also gives
H B = log ( 6 e N m ( B J + 1 ) ε ) ≤ C H ε . H_B=\log\left(\frac{6eNm(B_J+1)}{\varepsilon}\right)\le CH_\varepsilon. H B = log ( ε 6 e N m ( B J + 1 ) ) ≤ C H ε . Thus (36) bounds the depth and number of parameters in the shared feature network by C log ( e N ) H ε C\log(eN)H_\varepsilon C log ( e N ) H ε C m N H ε CmNH_\varepsilon C m N H ε (61) adds at most m N 1 mN_1 m N 1 (54) . Summing the branch and trunk parameter counts proves (55) .
Corollary 4.4. Under the assumptions of Theorem 4.3, let q ∈ [ 1 , ∞ ) q\in[1,\infty) q ∈ [ 1 , ∞ ) ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 )
sup u 0 ∈ G 0 ( E [ ∥ S ( u 0 , η ) − ∑ ℓ = 1 N 1 a ℓ ( u ⃗ 0 , c J ( η ) ) q ~ ℓ ∥ Y q ] ) 1 / q ≤ ε . (68) \sup_{u_0\in\mathcal{G}_0}\left(\mathbb{E}\left[\left\|S(u_0,\eta)-\sum_{\ell=1}^{N_1}a_\ell(\vec{u}_0,c_J(\eta))\widetilde{q}_\ell\right\|_Y^q\right]\right)^{1/q}\le\varepsilon. \tag*{(68)} u 0 ∈ G 0 sup ( E [ S ( u 0 , η ) − ℓ = 1 ∑ N 1 a ℓ ( u 0 , c J ( η )) q ℓ Y q ] ) 1/ q ≤ ε . (68) The measurement and trunk counts satisfy (49) –(51) , and the network depths and parameter counts satisfy (52) , (54) , and (55) , with
R = C q ′ log ( e / ε ) . R=C'_q\sqrt{\log(e/\varepsilon)}. R = C q ′ log ( e / ε ) . Here C q ′ ≥ 1 C'_q\ge1 C q ′ ≥ 1 N N N m m m B J B_J B J H ε H_\varepsilon H ε R R R q q q
Proof. Use the construction in Theorem 4.3, with the deterministic error on Ω R \Omega_R Ω R ε / 2 \varepsilon/2 ε /2 E θ E_\theta E θ
E θ 1 Ω R ≤ ε 2 . E_\theta1_{\Omega_R}\le\frac{\varepsilon}{2}. E θ 1 Ω R ≤ 2 ε . On Ω R c \Omega_R^c Ω R c K J , R K_{J,R} K J , R B J B_J B J
E θ ≤ C ( B J + 1 + ∥ η ∥ X 0 ) . E_\theta\le C(B_J+1+\|\eta\|_{X_0}). E θ ≤ C ( B J + 1 + ∥ η ∥ X 0 ) . For p 0 : = ( d + 4 ) / δ + 1 p_0:=(d+4)/\delta+1 p 0 := ( d + 4 ) / δ + 1 (65) and (32) imply B J + 1 ≤ C ε − p 0 ( 1 + R p 0 ) B_J+1\le C\varepsilon^{-p_0}(1+R^{p_0}) B J + 1 ≤ C ε − p 0 ( 1 + R p 0 ) B J + 1 ≥ 1 B_J+1\ge1 B J + 1 ≥ 1 ( B J + 1 + ∥ η ∥ X 0 ) q ≤ ( B J + 1 ) q ( 1 + ∥ η ∥ X 0 ) q (B_J+1+\|\eta\|_{X_0})^q\le(B_J+1)^q(1+\|\eta\|_{X_0})^q ( B J + 1 + ∥ η ∥ X 0 ) q ≤ ( B J + 1 ) q ( 1 + ∥ η ∥ X 0 ) q q q q ( 1 + R p 0 ) q ≤ 2 q ( 1 + R q p 0 ) (1+R^{p_0})^q\le2^q(1+R^{qp_0}) ( 1 + R p 0 ) q ≤ 2 q ( 1 + R q p 0 )
E [ E θ q 1 Ω R c ] ≤ C q ε − q p 0 ( 1 + R q p 0 ) e − c q R 2 . \mathbb{E}\left[E_\theta^q1_{\Omega_R^c}\right]\le C_q\varepsilon^{-qp_0}(1+R^{qp_0})e^{-c_qR^2}. E [ E θ q 1 Ω R c ] ≤ C q ε − q p 0 ( 1 + R q p 0 ) e − c q R 2 . Choose R = C q ′ log ( e / ε ) R=C'_q\sqrt{\log(e/\varepsilon)} R = C q ′ log ( e / ε ) C q ′ C'_q C q ′ ( ε / 2 ) q (\varepsilon/2)^q ( ε /2 ) q q q q 2 1 / q ε / 2 ≤ ε 2^{1/q}\varepsilon/2\leq\varepsilon 2 1/ q ε /2 ≤ ε q ≥ 1 q\geq1 q ≥ 1 (68) . With this choice of R R R Theorem 4.3 , with constants that may depend on q q q
Local approximation for stochastic Navier–Stokes equations
We consider the stochastic incompressible Navier–Stokes equations on T d \mathbb{T}^d T d d ≥ 2 d\geq2 d ≥ 2
The stochastic equation and solution spaces
Fix ν > 0 \nu>0 ν > 0
{ ∂ t u + ( u ⋅ ∇ ) u + ∇ p = ν Δ u + η , ( t , x ) ∈ ( 0 , T ] × T d , ∇ ⋅ u = 0 , u ( 0 ) = u 0 . (69) \begin{cases}
\partial_t u+(u\cdot\nabla)u+\nabla p=\nu\Delta u+\eta,\quad(t,x)\in(0,T]\times\mathbb{T}^d,\\
\nabla\cdot u=0,\quad u(0)=u_0. \tag*{(69)}
\end{cases} { ∂ t u + ( u ⋅ ∇ ) u + ∇ p = ν Δ u + η , ( t , x ) ∈ ( 0 , T ] × T d , ∇ ⋅ u = 0 , u ( 0 ) = u 0 . (69) Here η \eta η ( C g β 0 ( Λ T ) ) d (C_g^{\beta_0}(\Lambda_T))^d ( C g β 0 ( Λ T ) ) d D T D_T D T u 0 u_0 u 0 η \eta η P P P ϕ \phi ϕ
u ( t ) = e ν t Δ u 0 + K ν ϕ ( t ) − N ( u , u ) ( t ) , u(t)=e^{\nu t\Delta}u_0+K_\nu\phi(t)-N(u,u)(t), u ( t ) = e ν t Δ u 0 + K ν ϕ ( t ) − N ( u , u ) ( t ) , N ( u , v ) ( t ) : = ∫ 0 t e ν ( t − σ ) Δ P ∇ ⋅ ( u ⊗ v ) ( σ ) d σ . (70) N(u,v)(t):=\int_0^t e^{\nu(t-\sigma)\Delta}P\nabla\cdot(u\otimes v)(\sigma)\,\mathrm{d}\sigma. \tag*{(70)} N ( u , v ) ( t ) := ∫ 0 t e ν ( t − σ ) Δ P ∇ ⋅ ( u ⊗ v ) ( σ ) d σ . (70) We will apply this formulation both to realizations of η \eta η s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 )
C x s : = B ∞ , ∞ s ( T d ; R d ) , C g , x s : = C ∞ ( T d ; R d ) ‾ C x s . C_x^s:=B_{\infty,\infty}^s(\mathbb{T}^d;\mathbb{R}^d),\qquad C_{g,x}^s:=\overline{C^\infty(\mathbb{T}^d;\mathbb{R}^d)}^{C_x^s}. C x s := B ∞ , ∞ s ( T d ; R d ) , C g , x s := C ∞ ( T d ; R d ) C x s . The local-in-time solution space is
E τ : = { u ∈ C ( [ 0 , τ ] ; C x s ) : ∇ ⋅ u ( t ) = 0 , ∫ T d u ( t , x ) d x = 0 } , ∥ u ∥ E τ : = sup t ≤ τ ∥ u ( t ) ∥ C x s . E_\tau:=\left\{u\in C([0,\tau];C_x^s):\nabla\cdot u(t)=0,\ \int_{\mathbb{T}^d}u(t,x)\,\mathrm{d}x=0\right\},\qquad\|u\|_{E_\tau}:=\sup_{t\leq\tau}\|u(t)\|_{C_x^s}. E τ := { u ∈ C ([ 0 , τ ] ; C x s ) : ∇ ⋅ u ( t ) = 0 , ∫ T d u ( t , x ) d x = 0 } , ∥ u ∥ E τ := t ≤ τ sup ∥ u ( t ) ∥ C x s . We distinguish this space from the approximation space
Y τ : = C ( [ 0 , τ ] × T d ; R d ) , ∥ u ∥ Y τ : = sup ( t , x ) ∣ u ( t , x ) ∣ . Y_\tau:=C([0,\tau]\times\mathbb{T}^d;\mathbb{R}^d),\qquad\|u\|_{Y_\tau}:=\sup_{(t,x)}|u(t,x)|. Y τ := C ([ 0 , τ ] × T d ; R d ) , ∥ u ∥ Y τ := ( t , x ) sup ∣ u ( t , x ) ∣. Here C e m b C_{\mathrm{emb}} C emb ∥ u ∥ Y τ ≤ C e m b ∥ u ∥ E τ \|u\|_{Y_\tau}\leq C_{\mathrm{emb}}\|u\|_{E_\tau} ∥ u ∥ Y τ ≤ C emb ∥ u ∥ E τ τ ≤ T \tau\leq T τ ≤ T C x s C_x^s C x s C g , x s C_{g,x}^s C g , x s C x s C_x^s C x s C x s ˉ C_x^{\bar{s}} C x s ˉ s ˉ > s \bar{s}>s s ˉ > s C g , x s C_{g,x}^s C g , x s
Local existence and stability on a common interval
We present the estimates needed for the fixed-point argument. The spatial Littlewood–Paley norms are those of § 2.1 .
Lemma 5.1 (Product, projection, and semigroup estimates). Let 0 < s < 1 0 < s < 1 0 < s < 1 C s C_s C s d d d s s s
∥ f g ∥ C x s ≤ C s ∥ f ∥ C x s ∥ g ∥ C x s , \|fg\|_{C_x^s} \le C_s\|f\|_{C_x^s}\|g\|_{C_x^s}, ∥ f g ∥ C x s ≤ C s ∥ f ∥ C x s ∥ g ∥ C x s , ∥ P f ∥ C x s ≤ C s ∥ f ∥ C x s , \|Pf\|_{C_x^s} \le C_s\|f\|_{C_x^s}, ∥ P f ∥ C x s ≤ C s ∥ f ∥ C x s , ∥ e ν t Δ f ∥ C x s ≤ ∥ f ∥ C x s , t ≥ 0 , \|e^{\nu t\Delta}f\|_{C_x^s} \le\|f\|_{C_x^s}, \qquad t \ge0, ∥ e ν t Δ f ∥ C x s ≤ ∥ f ∥ C x s , t ≥ 0 , ∥ ( e ν h Δ − I ) g ∥ C x s ≤ C s ( ν h ) 1 / 2 ∥ g ∥ C x s + 1 , h > 0. \|(e^{\nu h\Delta}-I)g\|_{C_x^s} \le C_s(\nu h)^{1/2}\|g\|_{C_x^{s+1}}, \qquad h > 0. ∥ ( e ν h Δ − I ) g ∥ C x s ≤ C s ( ν h ) 1/2 ∥ g ∥ C x s + 1 , h > 0. The product estimate applies componentwise to tensor products. Moreover, t ↦ e ν t Δ f t \mapsto e^{\nu t\Delta}f t ↦ e ν t Δ f C ( [ 0 , T ] ; C x s ) C([0,T];C_x^s) C ([ 0 , T ] ; C x s ) f ∈ C g , x s f \in C_{g,x}^s f ∈ C g , x s
Proof. For 0 < s < 1 0 < s < 1 0 < s < 1 ∥ f ∥ ∞ + [ f ] s \|f\|_\infty+ [f]_s ∥ f ∥ ∞ + [ f ] s [ f ] s [f]_s [ f ] s [4 ] . Thus
[ f g ] s ≤ ∥ f ∥ ∞ [ g ] s + [ f ] s ∥ g ∥ ∞ , [fg]_s \le\|f\|_\infty[g]_s+[f]_s\|g\|_\infty, [ f g ] s ≤ ∥ f ∥ ∞ [ g ] s + [ f ] s ∥ g ∥ ∞ , which gives the product estimate. The symbol of P P P L 1 L^1 L 1 ∥ Δ j x P f ∥ ∞ ≤ C ∥ Δ j x f ∥ ∞ \|\Delta_j^x Pf\|_\infty\le C\|\Delta_j^x f\|_\infty ∥ Δ j x P f ∥ ∞ ≤ C ∥ Δ j x f ∥ ∞ P ( 0 ) P(0) P ( 0 )
The heat semigroup commutes with Δ j x \Delta_j^x Δ j x L ∞ L^\infty L ∞
∥ ( e ν h Δ − I ) Δ j x g ∥ ∞ ≤ C min { 1 , ν h 2 2 j } ∥ Δ j x g ∥ ∞ . \|(e^{\nu h\Delta}-I)\Delta_j^x g\|_\infty\le C\min\{1,\nu h2^{2j}\}\|\Delta_j^x g\|_\infty. ∥ ( e ν h Δ − I ) Δ j x g ∥ ∞ ≤ C min { 1 , ν h 2 2 j } ∥ Δ j x g ∥ ∞ . After multiplication by 2 j s 2^{js} 2 j s
C min { 2 − j , ν h 2 j } ∥ g ∥ C x s + 1 ≤ C ( ν h ) 1 / 2 ∥ g ∥ C x s + 1 . C\min\{2^{-j},\nu h2^j\}\|g\|_{C_x^{s+1}} \le C(\nu h)^{1/2}\|g\|_{C_x^{s+1}}. C min { 2 − j , ν h 2 j } ∥ g ∥ C x s + 1 ≤ C ( ν h ) 1/2 ∥ g ∥ C x s + 1 . The finitely many low modes satisfy the same estimate. Finally, if f ∈ C g , x s f \in C_{g,x}^s f ∈ C g , x s f n → f f_n \to f f n → f C x s C_x^s C x s
∥ ( e ν h Δ − I ) f ∥ C x s ≤ 2 ∥ f − f n ∥ C x s + ∥ ( e ν h Δ − I ) f n ∥ C x s . \|(e^{\nu h\Delta}-I)f\|_{C_x^s} \le2\|f-f_n\|_{C_x^s}+\|(e^{\nu h\Delta}-I)f_n\|_{C_x^s}. ∥ ( e ν h Δ − I ) f ∥ C x s ≤ 2∥ f − f n ∥ C x s + ∥ ( e ν h Δ − I ) f n ∥ C x s . First let h ↓ 0 h \downarrow0 h ↓ 0 n → ∞ n \to\infty n → ∞
The next embedding allows us to estimate the stochastic convolution in C ( [ 0 , T ] ; C x s ) C([0,T];C_x^s) C ([ 0 , T ] ; C x s ) α > s \alpha> s α > s C x s C_x^s C x s
Lemma 5.2 (From parabolic regularity to spatial paths). If α > s > 0 \alpha> s > 0 α > s > 0
C α ( D T ) ↪ C ( [ 0 , T ] ; C x s ) , ∥ v ∥ C ( [ 0 , T ] ; C x s ) ≤ C α , s ∥ v ∥ C α ( D T ) . C^\alpha(D_T) \hookrightarrow C([0,T];C_x^s), \qquad\|v\|_{C([0,T];C_x^s)} \le C_{\alpha,s}\|v\|_{C^\alpha(D_T)}. C α ( D T ) ↪ C ([ 0 , T ] ; C x s ) , ∥ v ∥ C ([ 0 , T ] ; C x s ) ≤ C α , s ∥ v ∥ C α ( D T ) . Proof. Extend v v v R × T d \mathbb{R}\times\mathbb{T}^d R × T d Δ j x Δ k s = 0 \Delta_j^x\Delta_k^s=0 Δ j x Δ k s = 0 k ≥ j − C k\ge j-C k ≥ j − C α > 0 \alpha>0 α > 0 t t t
∥ Δ j x v ( t ) ∥ ∞ ≤ C ∑ k ≥ j − C 2 − k α ∥ v ∥ C α ≤ C 2 − j α ∥ v ∥ C α . (71) \|\Delta_j^xv(t)\|_\infty\le C\sum_{k\ge j-C}2^{-k\alpha}\|v\|_{C^\alpha} \le C2^{-j\alpha}\|v\|_{C^\alpha}. \tag*{(71)} ∥ Δ j x v ( t ) ∥ ∞ ≤ C k ≥ j − C ∑ 2 − k α ∥ v ∥ C α ≤ C 2 − j α ∥ v ∥ C α . (71) This proves the norm estimate. To prove time continuity, set v L : = ∑ k ≤ L Δ k s v v_L := \sum_{k \le L} \Delta_k^s v v L := ∑ k ≤ L Δ k s v v L v_L v L
sup t ≤ T ∥ v ( t ) − v L ( t ) ∥ C x s ≤ C 2 − L ( α − s ) ∥ v ∥ C α . \sup_{t \le T} \|v(t)-v_L(t)\|_{C_x^s} \le C2^{-L(\alpha-s)}\|v\|_{C^\alpha}. t ≤ T sup ∥ v ( t ) − v L ( t ) ∥ C x s ≤ C 2 − L ( α − s ) ∥ v ∥ C α . Thus v L → v v_L \to v v L → v C ( [ 0 , T ] ; C x s ) C([0,T];C_x^s) C ([ 0 , T ] ; C x s )
Lemma 5.3 (Heat smoothing with one derivative). For 0 < s < 1 0 < s < 1 0 < s < 1 t > 0 t > 0 t > 0
∥ e ν t Δ P ∇ ⋅ G ∥ C x s ≤ C s ( ν t ) − 1 / 2 ∥ G ∥ C x s . \|e^{\nu t\Delta}P\nabla\cdot G\|_{C_x^s} \le C_s(\nu t)^{-1/2}\|G\|_{C_x^s}. ∥ e ν t Δ P ∇ ⋅ G ∥ C x s ≤ C s ( ν t ) − 1/2 ∥ G ∥ C x s . Proof. On a spatial annulus of frequency 2 j 2^j 2 j L 1 L^1 L 1 C 2 j e − c ν t 2 2 j C2^j e^{-c\nu t2^{2j}} C 2 j e − c ν t 2 2 j 2 j s 2^{js} 2 j s
sup r > 0 r e − c ν t r 2 ≤ C ( ν t ) − 1 / 2 \sup_{r>0} re^{-c\nu t r^2} \le C(\nu t)^{-1/2} r > 0 sup r e − c ν t r 2 ≤ C ( ν t ) − 1/2 gives the assertion on the annular blocks. The derivative annihilates the zero mode, and the finitely many remaining low modes obey the same bound. □
Lemma 5.4 (Bilinear estimate). For u , v ∈ E τ u,v \in E_\tau u , v ∈ E τ N ( u , v ) N(u,v) N ( u , v ) E τ E_\tau E τ t = 0 t=0 t = 0
∥ N ( u , v ) ∥ E τ ≤ C b τ / ν ∥ u ∥ E τ ∥ v ∥ E τ , \|N(u,v)\|_{E_\tau} \le C_b\sqrt{\tau/\nu}\|u\|_{E_\tau}\|v\|_{E_\tau}, ∥ N ( u , v ) ∥ E τ ≤ C b τ / ν ∥ u ∥ E τ ∥ v ∥ E τ , where C b C_b C b d d d s s s
Proof. Apply Lemma 5.3 and the product estimate to obtain
∥ N ( u , v ) ( t ) ∥ C x s ≤ C s 2 ∥ u ∥ E τ ∥ v ∥ E τ ∫ 0 t ( ν ( t − σ ) ) − 1 / 2 d σ = 2 C s 2 t / ν ∥ u ∥ E τ ∥ v ∥ E τ . \|N(u,v)(t)\|_{C_x^s} \le C_s^2\|u\|_{E_\tau}\|v\|_{E_\tau}\int_0^t(\nu(t-\sigma))^{-1/2}\,d\sigma
=2C_s^2\sqrt{t/\nu}\|u\|_{E_\tau}\|v\|_{E_\tau}. ∥ N ( u , v ) ( t ) ∥ C x s ≤ C s 2 ∥ u ∥ E τ ∥ v ∥ E τ ∫ 0 t ( ν ( t − σ ) ) − 1/2 d σ = 2 C s 2 t / ν ∥ u ∥ E τ ∥ v ∥ E τ . We may therefore take C b = 2 C s 2 C_b=2C_s^2 C b = 2 C s 2 h h h C x s C_x^s C x s C h / ν ∥ u ∥ E τ ∥ v ∥ E τ C\sqrt{h/\nu}\|u\|_{E_\tau}\|v\|_{E_\tau} C h / ν ∥ u ∥ E τ ∥ v ∥ E τ h ↓ 0 h\downarrow0 h ↓ 0
Proposition 5.5 (Local well-posedness and stability). Fix 0 < s < 1 0 < s < 1 0 < s < 1 ρ > 0 \rho> 0 ρ > 0 c ∗ > 0 c_* > 0 c ∗ > 0 d d d s s s 6 C b c ∗ ≤ 1 / 2 6C_b\sqrt{c_*}\le1/2 6 C b c ∗ ≤ 1/2
τ ∗ : = T ∧ c ∗ ν ρ 2 . \tau^* := T \wedge\frac{c_*\nu}{\rho^2}. τ ∗ := T ∧ ρ 2 c ∗ ν . Suppose that u 0 ∈ C g , x s u_0 \in C_{g,x}^s u 0 ∈ C g , x s K ν ϕ ∈ E T K_\nu\phi\in E_T K ν ϕ ∈ E T ( K ν ϕ ) ( 0 ) = 0 (K_\nu\phi)(0)=0 ( K ν ϕ ) ( 0 ) = 0
∥ u 0 ∥ C x s ≤ ρ , ∥ K ν ϕ ∥ E T ≤ ρ , (72) \|u_0\|_{C_x^s} \le\rho,\qquad\|K_\nu\phi\|_{E_T} \le\rho, \tag*{(72)} ∥ u 0 ∥ C x s ≤ ρ , ∥ K ν ϕ ∥ E T ≤ ρ , (72) then (69) has a unique solution in the closed ball { u ∈ E τ ∗ : ∥ u ∥ E τ ∗ ≤ 3 ρ } \{u\in E_{\tau^*}:\|u\|_{E_{\tau^*}}\le3\rho\} { u ∈ E τ ∗ : ∥ u ∥ E τ ∗ ≤ 3 ρ }
∥ u 1 − u 2 ∥ E τ ∗ ≤ 2 ( ∥ u 0 1 − u 0 2 ∥ C x s + ∥ K ν ( ϕ 1 − ϕ 2 ) ∥ E T ) . (73) \|u^1-u^2\|_{E_{\tau^*}} \le2\left(\|u_0^1-u_0^2\|_{C_x^s}+\|K_\nu(\phi^1-\phi^2)\|_{E_T}\right). \tag*{(73)} ∥ u 1 − u 2 ∥ E τ ∗ ≤ 2 ( ∥ u 0 1 − u 0 2 ∥ C x s + ∥ K ν ( ϕ 1 − ϕ 2 ) ∥ E T ) . (73) If s − 2 < β 1 < 0 s-2<\beta_1<0 s − 2 < β 1 < 0
∥ K ν ( ϕ 1 − ϕ 2 ) ∥ E T ≤ C ν , β 1 , s , T ∥ ϕ 1 − ϕ 2 ∥ C β 1 ( D T ) . (74) \|K_\nu(\phi^1-\phi^2)\|_{E_T} \le C_{\nu,\beta_1,s,T}\|\phi^1-\phi^2\|_{C^{\beta_1}(D_T)}. \tag*{(74)} ∥ K ν ( ϕ 1 − ϕ 2 ) ∥ E T ≤ C ν , β 1 , s , T ∥ ϕ 1 − ϕ 2 ∥ C β 1 ( D T ) . (74) Proof. Step 1: the invariant ball. Set a ( t ) : = e ν t Δ u 0 + K ν ϕ ( t ) a(t) := e^{\nu t\Delta}u_0 + K_\nu\phi(t) a ( t ) := e ν t Δ u 0 + K ν ϕ ( t ) T ( u ) : = a − N ( u , u ) T(u) := a - N(u,u) T ( u ) := a − N ( u , u ) a ∈ E τ ∗ a \in E_{\tau_*} a ∈ E τ ∗ ∥ a ∥ E τ ∗ ≤ 2 ρ \|a\|_{E_{\tau_*}} \le2\rho ∥ a ∥ E τ ∗ ≤ 2 ρ
∥ N ( u , v ) ∥ E τ ≤ C b τ / ν ∥ u ∥ E τ ∥ v ∥ E τ . (75) \|N(u,v)\|_{E_\tau} \le C_b\sqrt{\tau/\nu}\,\|u\|_{E_\tau}\|v\|_{E_\tau}. \tag*{(75)} ∥ N ( u , v ) ∥ E τ ≤ C b τ / ν ∥ u ∥ E τ ∥ v ∥ E τ . (75) Consequently, if ∥ u ∥ E τ ∗ ≤ 3 ρ \|u\|_{E_{\tau_*}} \le3\rho ∥ u ∥ E τ ∗ ≤ 3 ρ
∥ T ( u ) ∥ E τ ∗ ≤ 2 ρ + 9 C b ρ 2 τ ∗ / ν ≤ 2 ρ + 9 C b ρ c ∗ ≤ 11 4 ρ < 3 ρ . \|T(u)\|_{E_{\tau_*}} \le2\rho+ 9C_b\rho^2\sqrt{\tau_*/\nu} \le2\rho+ 9C_b\rho\sqrt{c_*} \le\frac{11}{4}\rho< 3\rho. ∥ T ( u ) ∥ E τ ∗ ≤ 2 ρ + 9 C b ρ 2 τ ∗ / ν ≤ 2 ρ + 9 C b ρ c ∗ ≤ 4 11 ρ < 3 ρ . Thus the free term is bounded by 2 ρ 2\rho 2 ρ 3 ρ / 4 3\rho/4 3 ρ /4 T T T 3 ρ 3\rho 3 ρ
Step 2: contraction and the choice of time. Bilinearity gives
N ( u , u ) − N ( v , v ) = N ( u , u − v ) + N ( u − v , v ) . N(u,u) - N(v,v) = N(u,u-v) + N(u-v,v). N ( u , u ) − N ( v , v ) = N ( u , u − v ) + N ( u − v , v ) . On the same ball, (73) therefore implies
∥ T ( u ) − T ( v ) ∥ E τ ∗ ≤ 6 C b ρ τ ∗ / ν ∥ u − v ∥ E τ ∗ ≤ 1 2 ∥ u − v ∥ E τ ∗ . \|T(u)-T(v)\|_{E_{\tau_*}} \le6C_b\rho\sqrt{\tau_*/\nu}\,\|u-v\|_{E_{\tau_*}} \le\frac{1}{2}\|u-v\|_{E_{\tau_*}}. ∥ T ( u ) − T ( v ) ∥ E τ ∗ ≤ 6 C b ρ τ ∗ / ν ∥ u − v ∥ E τ ∗ ≤ 2 1 ∥ u − v ∥ E τ ∗ . The condition τ ∗ ≤ c ∗ ν / ρ 2 \tau_* \le c_*\nu/\rho^2 τ ∗ ≤ c ∗ ν / ρ 2 1 / 2 1/2 1/2 t = 0 t=0 t = 0 u ( 0 ) = u 0 u(0)=u_0 u ( 0 ) = u 0
Step 3: stability with respect to both inputs. Subtract the two mild equations. Since both solutions lie in the 3 ρ 3\rho 3 ρ
∥ u 1 − u 2 ∥ E τ ∗ ≤ ∥ u 0 1 − u 0 2 ∥ C x β 1 + ∥ K ν ( ϕ 1 − ϕ 2 ) ∥ E T + 1 2 ∥ u 1 − u 2 ∥ E τ ∗ . \|u^1-u^2\|_{E_{\tau_*}} \le\|u_0^1-u_0^2\|_{C_x^{\beta_1}} + \|K_\nu(\phi^1-\phi^2)\|_{E_T} + \frac{1}{2}\|u^1-u^2\|_{E_{\tau_*}}. ∥ u 1 − u 2 ∥ E τ ∗ ≤ ∥ u 0 1 − u 0 2 ∥ C x β 1 + ∥ K ν ( ϕ 1 − ϕ 2 ) ∥ E T + 2 1 ∥ u 1 − u 2 ∥ E τ ∗ . Absorbing the last term proves (71). Finally, the viscous version of Lemma 2.4 gives K ν : C β 1 ( D T ) → C β 1 + 2 ( D T ) K_\nu: C^{\beta_1}(D_T) \to C^{\beta_1+2}(D_T) K ν : C β 1 ( D T ) → C β 1 + 2 ( D T ) ν \nu ν T T T α = β 1 + 2 > s \alpha=\beta_1+2>s α = β 1 + 2 > s
For fixed ρ \rho ρ
S l o c ( u 0 , ϕ ) : = u ∈ E τ ∗ \mathcal{S}^{\mathrm{loc}}(u_0,\phi) := u \in E_{\tau_*} S loc ( u 0 , ϕ ) := u ∈ E τ ∗ for this local solution. Its domain consists of the admissible data in Proposition 5.5; all subsequent evaluations use the same ρ \rho ρ τ ∗ \tau_* τ ∗
Fourier measurements and admissible coefficients
Fix 0 < s < s ˉ < 1 0<s<\bar{s}<1 0 < s < s ˉ < 1 M > 0 M>0 M > 0
G 0 N S E : = { u 0 ∈ C x s ˉ : ∇ ⋅ u 0 = 0 , ∫ T d u 0 = 0 , ∥ u 0 ∥ C x s ˉ ≤ M } . \mathcal{G}^{\mathrm{NSE}}_0 := \left\{u_0 \in C_x^{\bar{s}} : \nabla\cdot u_0=0,\ \int_{\mathbb{T}^d}u_0=0,\ \|u_0\|_{C_x^{\bar{s}}}\le M\right\}. G 0 NSE := { u 0 ∈ C x s ˉ : ∇ ⋅ u 0 = 0 , ∫ T d u 0 = 0 , ∥ u 0 ∥ C x s ˉ ≤ M } . The initial velocity is represented by its retained spatial Fourier coefficients and reconstructed using the cutoff Π J 0 x \Pi^x_{J_0} Π J 0 x
The measurement maps act on each of the d d d
D 0 : = d N 0 , N 0 : = ∣ Λ J 0 x ∣ , D J : = d N J , N J : = ∣ Λ J ∣ . D_0 := dN_0,\qquad N_0 := |\Lambda_{J_0}^x|,\qquad D_J := dN_J,\qquad N_J := |\Lambda_J|. D 0 := d N 0 , N 0 := ∣ Λ J 0 x ∣ , D J := d N J , N J := ∣ Λ J ∣. Thus N 0 N_0 N 0 N J N_J N J D 0 D_0 D 0 D J D_J D J d d d D 0 ≍ 2 J 0 d D_0 \asymp2^{J_0d} D 0 ≍ 2 J 0 d D J ≍ 2 J ( d + 2 ) D_J \asymp2^{J(d+2)} D J ≍ 2 J ( d + 2 )
Lemma 5.6 (Spatial Fourier measurements). Let
c J 0 x ( u 0 ) : = ( ( u 0 , a , e λ ) 1 ≤ a ≤ d , λ ∈ Λ J 0 x ∈ R D 0 , ( Rec J 0 x ( v ) ) a : = ∑ λ ∈ Λ J 0 x m J 0 x ( λ ) v a , λ e λ . c^{x}_{J_0}(u_0):=((u_{0,a},e_\lambda)_{1\leq a\leq d,\ \lambda\in\Lambda^{x}_{J_0}}\in\mathbb{R}^{D_0},\qquad(\operatorname{Rec}^{x}_{J_0}(v))_a:=\sum_{\lambda\in\Lambda^{x}_{J_0}}m^{x}_{J_0}(\lambda)v_{a,\lambda}e_\lambda. c J 0 x ( u 0 ) := (( u 0 , a , e λ ) 1 ≤ a ≤ d , λ ∈ Λ J 0 x ∈ R D 0 , ( Rec J 0 x ( v ) ) a := λ ∈ Λ J 0 x ∑ m J 0 x ( λ ) v a , λ e λ . Then Rec J 0 x ( c J 0 x ( u 0 ) ) = Π J 0 x u 0 \operatorname{Rec}^{x}_{J_0}(c^{x}_{J_0}(u_0))=\Pi^{x}_{J_0}u_0 Rec J 0 x ( c J 0 x ( u 0 )) = Π J 0 x u 0
∥ ( I − Π J 0 x ) u 0 ∥ C x s ˉ ≤ C 2 − J 0 ( s ˉ − s ) M , u 0 ∈ G 0 N S E , (76) \left\|(I-\Pi^{x}_{J_0})u_0\right\|_{C^{\bar{s}}_x}\leq C2^{-J_0(\bar{s}-s)}M,\qquad u_0\in\mathcal{G}^{\mathrm{NSE}}_0, \tag*{(76)} ( I − Π J 0 x ) u 0 C x s ˉ ≤ C 2 − J 0 ( s ˉ − s ) M , u 0 ∈ G 0 NSE , (76) ∥ Rec J 0 x ( v ) − Rec J 0 x ( v ′ ) ∥ C x s ≤ C 2 J 0 s D 0 1 / 2 ∣ v − v ′ ∣ 2 . (77) \left\|\operatorname{Rec}^{x}_{J_0}(v)-\operatorname{Rec}^{x}_{J_0}(v')\right\|_{C^s_x}\leq C2^{J_0s}D_0^{1/2}|v-v'|_2. \tag*{(77)} Rec J 0 x ( v ) − Rec J 0 x ( v ′ ) C x s ≤ C 2 J 0 s D 0 1/2 ∣ v − v ′ ∣ 2 . (77) Moreover, uniformly over J 0 ≥ 0 J_0\geq0 J 0 ≥ 0
∣ c J 0 x ( u 0 ) ∣ ∞ ≤ C M , ∥ u 0 ∥ C x s + ∥ Π J 0 x u 0 ∥ C x s ≤ C M . (78) |c^{x}_{J_0}(u_0)|_\infty\leq CM,\qquad\|u_0\|_{C^s_x}+\|\Pi^{x}_{J_0}u_0\|_{C^s_x}\leq CM. \tag*{(78)} ∣ c J 0 x ( u 0 ) ∣ ∞ ≤ C M , ∥ u 0 ∥ C x s + ∥ Π J 0 x u 0 ∥ C x s ≤ C M . (78) Both u 0 u_0 u 0 C g , x s C^s_{g,x} C g , x s
Proof. The smooth cutoff is uniformly bounded on every spatial Besov space. The multiplier I − Π J 0 x I-\Pi^{x}_{J_0} I − Π J 0 x 2 J 0 2^{J_0} 2 J 0
2 j s ∥ Δ j x ( I − Π J 0 x ) u 0 ∥ ∞ ≤ C 2 − j ( s ˉ − s ) ∥ u 0 ∥ C x s ˉ , j ≥ J 0 − C . 2^{js}\|\Delta^x_j(I-\Pi^{x}_{J_0})u_0\|_\infty\leq C2^{-j(\bar{s}-s)}\|u_0\|_{C^{\bar{s}}_x},\qquad j\geq J_0-C. 2 j s ∥ Δ j x ( I − Π J 0 x ) u 0 ∥ ∞ ≤ C 2 − j ( s ˉ − s ) ∥ u 0 ∥ C x s ˉ , j ≥ J 0 − C . Taking the supremum proves (74). A retained trigonometric mode has frequency at most C 2 J 0 C2^{J_0} C 2 J 0 C x s C^s_x C x s C 2 J 0 s C2^{J_0s} C 2 J 0 s ∥ u 0 ∥ ∞ ∥ e λ ∥ L 1 \|u_0\|_\infty\|e_\lambda\|_{L^1} ∥ u 0 ∥ ∞ ∥ e λ ∥ L 1 ∥ e λ ∥ L 1 \|e_\lambda\|_{L^1} ∥ e λ ∥ L 1 C x s ˉ ↪ C x s C^{\bar{s}}_x\hookrightarrow C^s_x C x s ˉ ↪ C x s u 0 u_0 u 0 C x s C^s_x C x s
For the forcing, use c J c_J c J Rec J \operatorname{Rec}_J Rec J R D J \mathbb{R}^{D_J} R D J s − 2 < β 1 < β 0 < 0 s-2<\beta_1<\beta_0<0 s − 2 < β 1 < β 0 < 0
X 0 = ( C g β 0 ( Λ T ) ) d , X 1 = ( C β 1 ( D T ) ) d , δ : = β 0 − β 1 > 0. X_0=(C^{\beta_0}_g(\Lambda_T))^d,\qquad X_1=(C^{\beta_1}(D_T))^d,\qquad\delta:=\beta_0-\beta_1>0. X 0 = ( C g β 0 ( Λ T ) ) d , X 1 = ( C β 1 ( D T ) ) d , δ := β 0 − β 1 > 0. The estimates of Lemmas 3.2 and 3.3 hold componentwise. In particular,
∥ ( I − Π J ) ζ ∥ X 1 ≤ C 2 − J δ ∥ ζ ∥ X 0 , (79) \|(I-\Pi_J)\zeta\|_{X_1}\leq C2^{-J\delta}\|\zeta\|_{X_0}, \tag*{(79)} ∥ ( I − Π J ) ζ ∥ X 1 ≤ C 2 − J δ ∥ ζ ∥ X 0 , (79) |c_J(\zeta)|_\infty\leq C2^{-J\beta_0}|\zeta|_{X_0},
{#eq:78}
∥ Rec J ( c ) − Rec J ( c ′ ) ∥ X 1 ≤ C D J 1 / 2 ∣ c − c ′ ∣ 2 . \|\operatorname{Rec}_J(c)-\operatorname{Rec}_J(c')\|_{X_1}\leq CD_J^{1/2}|c-c'|_2. ∥ Rec J ( c ) − Rec J ( c ′ ) ∥ X 1 ≤ C D J 1/2 ∣ c − c ′ ∣ 2 . The finite-dimensional domain must contain the measurement vectors and must also allow the local solution to be evaluated at every center of the partition of unity. Let V J 0 ⊂ R D 0 \mathcal{V}_{J_0}\subset\mathbb{R}^{D_0} V J 0 ⊂ R D 0 W J ⊂ R D J \mathcal{W}_J\subset\mathbb{R}^{D_J} W J ⊂ R D J
v ∈ V J 0 ⟺ ∇ ⋅ Rec J 0 x ( v ) = 0 , ∫ T d Rec J 0 x ( v ) = 0 , v\in\mathcal{V}_{J_0}\quad\Longleftrightarrow\quad\nabla\cdot\operatorname{Rec}^{x}_{J_0}(v)=0,\qquad\int_{\mathbb{T}^d}\operatorname{Rec}^{x}_{J_0}(v)=0, v ∈ V J 0 ⟺ ∇ ⋅ Rec J 0 x ( v ) = 0 , ∫ T d Rec J 0 x ( v ) = 0 , c ∈ W J ⟺ ∇ ⋅ Rec J ( c ) = 0 , ∫ T d Rec J ( c ) ( t , x ) d x = 0 for all t . c\in\mathcal{W}_J\quad\Longleftrightarrow\quad\nabla\cdot\operatorname{Rec}_J(c)=0,\qquad\int_{\mathbb{T}^d}\operatorname{Rec}_J(c)(t,x)\,dx=0\quad\text{for all }t. c ∈ W J ⟺ ∇ ⋅ Rec J ( c ) = 0 , ∫ T d Rec J ( c ) ( t , x ) d x = 0 for all t . These are finite linear constraints: at each nonzero spatial frequency, the vector coefficient is orthogonal to that frequency, and the spatial zero modes vanish.
Choose fixed constants C 0 , C 1 C_0,C_1 C 0 , C 1 R J : = C 1 2 − J β 0 ρ R_J:=C_1 2^{-J\beta_0}\rho R J := C 1 2 − J β 0 ρ
K J , J 0 , ρ N S E : = { ( v , c ) ∈ ( [ − C 0 M , C 0 M ] D 0 ∩ V J 0 ) × ( [ − R J , R J ] D J ∩ W J ) : ∥ Rec J 0 x ( v ) ∥ C x s ≤ ρ , ∥ K ν Rec J ( c ) ∥ E T ≤ ρ } . \mathcal{K}^{\mathrm{NSE}}_{J,J_0,\rho}:=\left\{(v,c)\in\left([-C_0M,C_0M]^{D_0}\cap\mathcal{V}_{J_0}\right)\times\left([-R_J,R_J]^{D_J}\cap\mathcal{W}_J\right):
\left\|\operatorname{Rec}^{x}_{J_0}(v)\right\|_{C^s_x}\leq\rho,\ \left\|K_\nu\operatorname{Rec}_J(c)\right\|_{E_T}\leq\rho\right\}. K J , J 0 , ρ NSE := { ( v , c ) ∈ ( [ − C 0 M , C 0 M ] D 0 ∩ V J 0 ) × ( [ − R J , R J ] D J ∩ W J ) : Rec J 0 x ( v ) C x s ≤ ρ , ∥ K ν Rec J ( c ) ∥ E T ≤ ρ } . This set is compact: the linear constraints and the two norm constraints are closed, and the enclosing box is compact. The reconstructed fields are smooth, so the initial trace requirements of Proposition 5.5 also hold. Hence every point of this set, not only measurement vectors arising from the random forcing, defines a solution on the common interval [ 0 , τ ∗ ] [0,\tau^*] [ 0 , τ ∗ ]
The approximation theorem
The forcing is the centered Gaussian random variable of Assumption 2.2, interpreted componentwise, with the additional divergence and mean constraints. The requirement β 0 + 2 > s \beta_0+2>s β 0 + 2 > s C x s C_x^s C x s ( t , x ) (t,x) ( t , x )
Theorem 5.8 (High-probability local operator approximation). Let d ≥ 2 d\ge2 d ≥ 2 ν , T , M > 0 \nu,T,M>0 ν , T , M > 0 0 < s < s ˉ < 1 0<s<\bar{s}<1 0 < s < s ˉ < 1 η \eta η X 0 X_0 X 0
s − 2 < β 0 < 0. s-2<\beta_0<0. s − 2 < β 0 < 0. Fix β 1 ∈ ( s − 2 , β 0 ) \beta_1\in(s-2,\beta_0) β 1 ∈ ( s − 2 , β 0 ) δ = β 0 − β 1 \delta=\beta_0-\beta_1 δ = β 0 − β 1 C 0 , C ν ≥ 1 C_0,C_\nu\ge1 C 0 , C ν ≥ 1 ρ ≥ C 0 M \rho\ge C_0M ρ ≥ C 0 M τ ∗ = T ∧ c ∗ ν / ρ 2 \tau^*=T\wedge c_*\nu/\rho^2 τ ∗ = T ∧ c ∗ ν / ρ 2
Set
G ρ : = { C ν ∥ η ∥ X 0 ≤ ρ } , P ( G ρ c ) ≤ C e − c ρ 2 . G_\rho:=\{C_\nu\|\eta\|_{X_0}\le\rho\}, \qquad\mathbb{P}(G_\rho^c)\le Ce^{-c\rho^2}. G ρ := { C ν ∥ η ∥ X 0 ≤ ρ } , P ( G ρ c ) ≤ C e − c ρ 2 . For each ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) J , J 0 J,J_0 J , J 0 m m m a ~ k \widetilde{a}_k a k q ~ k \widetilde{q}_k q k
Γ θ N S E ( u 0 , ζ ) ( t , x ) : = ∑ k = 1 m a ~ k ( c J 0 x ( u 0 ) , c J ( ζ ) ) q ~ k ( t , x ) (80) \Gamma_\theta^{\mathrm{NSE}}(u_0,\zeta)(t,x):=\sum_{k=1}^{m}\widetilde{a}_k(c_{J_0}^{x}(u_0),c_J(\zeta))\widetilde{q}_k(t,x) \tag*{(80)} Γ θ NSE ( u 0 , ζ ) ( t , x ) := k = 1 ∑ m a k ( c J 0 x ( u 0 ) , c J ( ζ )) q k ( t , x ) (80) satisfies the deterministic bound
sup u 0 ∈ G N S E ζ ∈ X 0 , C ν ∥ ζ ∥ X 0 ≤ ρ ∇ ⋅ ζ = 0 , ∫ T T ζ d ξ = 0 ∥ S l o c ( u 0 , ζ ) − Γ θ N S E ( u 0 , ζ ) ∥ Y τ ∗ ≤ ε . (81) \sup_{\substack{u_0\in\mathcal{G}^{\mathrm{NSE}}\\ \zeta\in X_0,\ C_\nu\|\zeta\|_{X_0}\le\rho\\ \nabla\cdot\zeta=0,\ \int_{\mathbb{T}_T}\zeta\,d\xi=0}}
\left\|\mathcal{S}^{\mathrm{loc}}(u_0,\zeta)-\Gamma_\theta^{\mathrm{NSE}}(u_0,\zeta)\right\|_{Y_{\tau^*}}\le\varepsilon. \tag*{(81)} u 0 ∈ G NSE ζ ∈ X 0 , C ν ∥ ζ ∥ X 0 ≤ ρ ∇ ⋅ ζ = 0 , ∫ T T ζ d ξ = 0 sup S loc ( u 0 , ζ ) − Γ θ NSE ( u 0 , ζ ) Y τ ∗ ≤ ε . (81) In particular,
sup u 0 ∈ G 0 N S E E [ 1 G ρ ∥ S l o c ( u 0 , η ) − Γ θ N S E ( u 0 , η ) ∥ Y τ ∗ ] ≤ ε , (82) \sup_{u_0\in\mathcal{G}_0^{\mathrm{NSE}}}\mathbb{E}\left[1_{G_\rho}\left\|\mathcal{S}^{\mathrm{loc}}(u_0,\eta)-\Gamma_\theta^{\mathrm{NSE}}(u_0,\eta)\right\|_{Y_{\tau^*}}\right]\leq\varepsilon, \tag*{(82)} u 0 ∈ G 0 NSE sup E [ 1 G ρ S loc ( u 0 , η ) − Γ θ NSE ( u 0 , η ) Y τ ∗ ] ≤ ε , (82) where the error inside the expectation is defined to be zero off G ρ G_\rho G ρ
D J ≤ C ( 1 + ρ ε ) ( d + 2 ) / δ , D 0 ≤ C ( 1 + M ε ) d / ( s ˉ − s ) . (83) D_J\leq C\left(1+\frac{\rho}{\varepsilon}\right)^{(d+2)/\delta},\qquad D_0\leq C\left(1+\frac{M}{\varepsilon}\right)^{d/(\bar{s}-s)}. \tag*{(83)} D J ≤ C ( 1 + ε ρ ) ( d + 2 ) / δ , D 0 ≤ C ( 1 + ε M ) d / ( s ˉ − s ) . (83) The constants are independent of ρ \rho ρ ε \varepsilon ε η \eta η [ 0 , τ ∗ ] [0,\tau^*] [ 0 , τ ∗ ] ε \varepsilon ε
Proof. We first choose an interval on which the original and truncated solutions are defined. We then bound each of four errors by ε / 4 \varepsilon/4 ε /4
First of all, the Schauder estimate, Lemma 5.2, and uniform boundedness of the smooth cutoffs give
∥ K ν ζ ∥ E T + sup J ≥ 0 ∥ K ν Π J ζ ∥ E T ≤ C ν ∥ ζ ∥ X 0 . (84) \left\|K_\nu\zeta\right\|_{E_T}+\sup_{J\geq0}\left\|K_\nu\Pi_J\zeta\right\|_{E_T}\leq C_\nu\left\|\zeta\right\|_{X_0}. \tag*{(84)} ∥ K ν ζ ∥ E T + J ≥ 0 sup ∥ K ν Π J ζ ∥ E T ≤ C ν ∥ ζ ∥ X 0 . (84) Indeed, the Schauder estimate yields parabolic regularity β 0 + 2 \beta_0+2 β 0 + 2 β 0 + 2 > s \beta_0+2>s β 0 + 2 > s E T E_T E T ζ ∈ X 0 \zeta\in X_0 ζ ∈ X 0 E T E_T E T C ν C_\nu C ν C ν ≥ 1 C_\nu\geq1 C ν ≥ 1 C 0 C_0 C 0
∥ u 0 ∥ C x s ˉ ≤ ρ , sup J 0 ≥ 0 ∥ Π J 0 x u 0 ∥ C x s ˉ ≤ ρ \left\|u_0\right\|_{C_x^{\bar{s}}}\leq\rho,\qquad\sup_{J_0\geq0}\left\|\Pi_{J_0}^x u_0\right\|_{C_x^{\bar{s}}}\leq\rho ∥ u 0 ∥ C x s ˉ ≤ ρ , J 0 ≥ 0 sup Π J 0 x u 0 C x s ˉ ≤ ρ whenever u 0 ∈ G 0 N S E u_0\in\mathcal{G}_0^{\mathrm{NSE}} u 0 ∈ G 0 NSE ρ ≥ C 0 M \rho\geq C_0M ρ ≥ C 0 M
For the remainder of the deterministic argument, we fix any u 0 u_0 u 0 ζ \zeta ζ (81) . Then, estimate (84) shows that ζ \zeta ζ Π J ζ \Pi_J\zeta Π J ζ (72) .
Thus Proposition 5.5 gives solutions on the common interval [ 0 , τ ∗ ] [0,\tau^*] [ 0 , τ ∗ ]
∥ S l o c ( u 0 , ζ ) ∥ E τ ∗ ≤ 3 ρ , \left\|\mathcal{S}^{\mathrm{loc}}(u_0,\zeta)\right\|_{E_{\tau^*}}\leq3\rho, S loc ( u 0 , ζ ) E τ ∗ ≤ 3 ρ , ∥ S l o c ( u 0 , Π J ζ ) ∥ E τ ∗ ≤ 3 ρ , \left\|\mathcal{S}^{\mathrm{loc}}(u_0,\Pi_J\zeta)\right\|_{E_{\tau^*}}\leq3\rho, S loc ( u 0 , Π J ζ ) E τ ∗ ≤ 3 ρ , ∥ S l o c ( Π J 0 x u 0 , Π J ζ ) ∥ E τ ∗ ≤ 3 ρ . \left\|\mathcal{S}^{\mathrm{loc}}(\Pi_{J_0}^x u_0,\Pi_J\zeta)\right\|_{E_{\tau^*}}\leq3\rho. S loc ( Π J 0 x u 0 , Π J ζ ) E τ ∗ ≤ 3 ρ . Since these bounds are uniform in J J J J 0 J_0 J 0
Next, by (73) , (74) , and (79) , we see
∥ S l o c ( u 0 , ζ ) − S l o c ( u 0 , Π J ζ ) ∥ Y τ ∗ ≤ 2 C e m b ∥ K ν ( I − Π J ) ζ ∥ E T ≤ C ∥ ( I − Π J ) ζ ∥ X 1 ≤ C f 2 − J δ ρ . \begin{aligned}
\left\|\mathcal{S}^{\mathrm{loc}}(u_0,\zeta)-\mathcal{S}^{\mathrm{loc}}(u_0,\Pi_J\zeta)\right\|_{Y_{\tau^*}}
&\leq2C_{\mathrm{emb}}\left\|K_\nu(I-\Pi_J)\zeta\right\|_{E_T}\\
&\leq C\left\|(I-\Pi_J)\zeta\right\|_{X_1}\leq Cf2^{-J\delta}\rho.
\end{aligned} S loc ( u 0 , ζ ) − S loc ( u 0 , Π J ζ ) Y τ ∗ ≤ 2 C emb ∥ K ν ( I − Π J ) ζ ∥ E T ≤ C ∥ ( I − Π J ) ζ ∥ X 1 ≤ C f 2 − J δ ρ . Here, β 1 < β 0 \beta_1<\beta_0 β 1 < β 0 β 1 + 2 > s \beta_1+2>s β 1 + 2 > s E T E_T E T E τ ∗ ↪ Y τ ∗ E_{\tau^*}\hookrightarrow Y_{\tau^*} E τ ∗ ↪ Y τ ∗ ∥ u 0 ∥ C x s ˉ ≤ M \left\|u_0\right\|_{C_x^{\bar{s}}}\leq M ∥ u 0 ∥ C x s ˉ ≤ M s ˉ > s \bar{s}>s s ˉ > s
∥ S l o c ( u 0 , Π J ζ ) − S l o c ( Π J 0 x u 0 , Π J ζ ) ∥ Y τ ∗ ≤ 2 C e m b ∥ ( I − Π J 0 x ) u 0 ∥ C x s ≤ C i 2 − J 0 ( s ˉ − s ) M . \begin{aligned}
\left\|\mathcal{S}^{\mathrm{loc}}(u_0,\Pi_J\zeta)-\mathcal{S}^{\mathrm{loc}}(\Pi_{J_0}^x u_0,\Pi_J\zeta)\right\|_{Y_{\tau^*}}
&\leq2C_{\mathrm{emb}}\left\|(I-\Pi_{J_0}^x)u_0\right\|_{C_x^s}\\
&\leq Ci2^{-J_0(\bar{s}-s)}M.
\end{aligned} S loc ( u 0 , Π J ζ ) − S loc ( Π J 0 x u 0 , Π J ζ ) Y τ ∗ ≤ 2 C emb ( I − Π J 0 x ) u 0 C x s ≤ C i 2 − J 0 ( s ˉ − s ) M . Choose the smallest nonnegative integers J , J 0 J,J_0 J , J 0
2 J δ ≥ 4 C f ρ ε , 2 J 0 ( s ˉ − s ) ≥ 4 C i M ε . 2^{J\delta}\geq\frac{4C_f\rho}{\varepsilon},\qquad2^{J_0(\bar{s}-s)}\geq\frac{4C_iM}{\varepsilon}. 2 J δ ≥ ε 4 C f ρ , 2 J 0 ( s ˉ − s ) ≥ ε 4 C i M . Both errors are then at most ε / 4 \varepsilon/4 ε /4 J J J J 0 J_0 J 0
2 J ≤ 2 max { 1 , ( 4 C f ρ / ε ) 1 / δ } 2^J \le2\max\left\{1,\left(4C_{\mathrm{f}}\rho/\varepsilon\right)^{1/\delta}\right\} 2 J ≤ 2 max { 1 , ( 4 C f ρ / ε ) 1/ δ } and similarly for J 0 J_0 J 0 (83) .
We abbreviate K : = K J , J 0 , ρ N S E \mathcal{K}:=\mathcal{K}^{\mathrm{NSE}}_{J,J_0,\rho} K := K J , J 0 , ρ NSE
Γ J , J 0 N S E ( v , c ) : = S l o c ( Rec J 0 x ( v ) , Rec J ( c ) ) , ( v , c ) ∈ K . \Gamma^{\mathrm{NSE}}_{J,J_0}(v,c):=\mathcal{S}^{\mathrm{loc}}\left(\operatorname{Rec}^{x}_{J_0}(v),\operatorname{Rec}_{J}(c)\right),\qquad(v,c)\in\mathcal{K}. Γ J , J 0 NSE ( v , c ) := S loc ( Rec J 0 x ( v ) , Rec J ( c ) ) , ( v , c ) ∈ K . Every point of K \mathcal{K} K z = ( c J 0 x ( u 0 ) , c J ( ζ ) ) z=(c^{x}_{J_0}(u_0),c_J(\zeta)) z = ( c J 0 x ( u 0 ) , c J ( ζ )) K \mathcal{K} K (84) and (78) ; the coefficient bounds follow from (78) and (78), using ∥ ζ ∥ X 0 ≤ ρ / C ν ≤ ρ \|\zeta\|_{X_0}\le\rho/C_\nu\le\rho ∥ ζ ∥ X 0 ≤ ρ / C ν ≤ ρ
Γ J , J 0 N S E ( z ) = S l o c ( Π J 0 x u 0 , Π J ζ ) . (85) \Gamma^{\mathrm{NSE}}_{J,J_0}(z)=\mathcal{S}^{\mathrm{loc}}\left(\Pi^{x}_{J_0}u_0,\Pi_J\zeta\right). \tag*{(85)} Γ J , J 0 NSE ( z ) = S loc ( Π J 0 x u 0 , Π J ζ ) . (85) For z = ( v , c ) z=(v,c) z = ( v , c ) z ′ = ( v ′ , c ′ ) ∈ K z'=(v',c')\in\mathcal{K} z ′ = ( v ′ , c ′ ) ∈ K
∥ Γ J , J 0 N S E ( z ) − Γ J , J 0 N S E ( z ′ ) ∥ Y τ ∗ ≤ C ( 2 2 J 0 s D 0 1 / 2 ∣ v − v ′ ∣ 2 + D J 1 / 2 ∣ c − c ′ ∣ 2 ) ≤ L ∣ z − z ′ ∣ 2 , \left\|\Gamma^{\mathrm{NSE}}_{J,J_0}(z)-\Gamma^{\mathrm{NSE}}_{J,J_0}(z')\right\|_{Y_{\tau_*}}
\le C\left(2^{2J_0s}D_0^{1/2}|v-v'|_2+D_J^{1/2}|c-c'|_2\right)
\le L|z-z'|_2, Γ J , J 0 NSE ( z ) − Γ J , J 0 NSE ( z ′ ) Y τ ∗ ≤ C ( 2 2 J 0 s D 0 1/2 ∣ v − v ′ ∣ 2 + D J 1/2 ∣ c − c ′ ∣ 2 ) ≤ L ∣ z − z ′ ∣ 2 , where we take
L : = max { 1 , C ( 2 2 J 0 s D 0 + D J ) 1 / 2 } . L:=\max\left\{1,C\left(2^{2J_0s}D_0+D_J\right)^{1/2}\right\}. L := max { 1 , C ( 2 2 J 0 s D 0 + D J ) 1/2 } . Next, we put
r : = ε 4 L (86) r:=\frac{\varepsilon}{4L} \tag*{(86)} r := 4 L ε (86) and choose an r / 2 r/2 r /2 z 1 , … , z m z^1,\ldots,z^m z 1 , … , z m K \mathcal{K} K K \mathcal{K} K Lemma 3.10 , we get
h k ( z ) : = ( 1 − ∣ z − z k ∣ 2 r ) + , ρ k ( z ) : = h k ( z ) ∑ j = 1 m h j ( z ) , z ∈ K . h_k(z):=\left(1-\frac{|z-z^k|_2}{r}\right)_+,\qquad
\rho_k(z):=\frac{h_k(z)}{\sum_{j=1}^{m}h_j(z)},\qquad z\in\mathcal{K}. h k ( z ) := ( 1 − r ∣ z − z k ∣ 2 ) + , ρ k ( z ) := ∑ j = 1 m h j ( z ) h k ( z ) , z ∈ K . The denominator is at least 1 / 2 1/2 1/2 r / 2 r/2 r /2 ρ k ≥ 0 \rho_k\ge0 ρ k ≥ 0 ∑ k ρ k = 1 \sum_k\rho_k=1 ∑ k ρ k = 1 Q k : = Γ J , J 0 N S E ( z k ) Q_k:=\Gamma^{\mathrm{NSE}}_{J,J_0}(z^k) Q k := Γ J , J 0 NSE ( z k )
A ( z ) ( t , x ) : = ∑ k = 1 m ρ k ( z ) Q k ( t , x ) . A(z)(t,x):=\sum_{k=1}^{m}\rho_k(z)Q_k(t,x). A ( z ) ( t , x ) := k = 1 ∑ m ρ k ( z ) Q k ( t , x ) . Each Q k Q_k Q k K \mathcal{K} K ρ k ( z ) > 0 \rho_k(z)>0 ρ k ( z ) > 0 ∣ z − z k ∣ 2 < r |z-z^k|_2<r ∣ z − z k ∣ 2 < r
∥ Γ J , J 0 N S E ( z ) − A ( z ) ∥ Y τ ∗ ≤ ∑ k ρ k ( z ) ∥ Γ J , J 0 N S E ( z ) − Q k ∥ Y τ ∗ ≤ ∑ k ρ k ( z ) L r = L r = ε 4 . \begin{aligned}
\left\|\Gamma^{\mathrm{NSE}}_{J,J_0}(z)-A(z)\right\|_{Y_{\tau_*}}
\le\sum_k\rho_k(z)\left\|\Gamma^{\mathrm{NSE}}_{J,J_0}(z)-Q_k\right\|_{Y_{\tau_*}} \\
&\le\sum_k\rho_k(z)Lr=Lr=\frac{\varepsilon}{4}.
\end{aligned} Γ J , J 0 NSE ( z ) − A ( z ) Y τ ∗ ≤ k ∑ ρ k ( z ) Γ J , J 0 NSE ( z ) − Q k Y τ ∗ ≤ k ∑ ρ k ( z ) L r = L r = 4 ε . The choice (86) therefore gives the required partition-of-unity error. Each reference solution satisfies
∥ Q k ∥ Y τ ∗ ≤ 3 C e m b ρ = : B . \|Q_k\|_{Y_{\tau_*}}\le3C_{\mathrm{emb}}\rho=:B. ∥ Q k ∥ Y τ ∗ ≤ 3 C emb ρ =: B . For δ T , δ B > 0 \delta_T,\delta_B>0 δ T , δ B > 0 q ~ k \widetilde q_k q k a ~ k \widetilde a_k a k
∥ Q k − q ~ k ∥ Y τ ∗ ≤ δ T , sup z ∈ K ∣ ρ k ( z ) − a ~ k ( z ) ∣ ≤ δ B . (87) \|Q_k-\widetilde q_k\|_{Y_{\tau_*}}\le\delta_T,\qquad
\sup_{z\in\mathcal{K}}|\rho_k(z)-\widetilde a_k(z)|\le\delta_B. \tag*{(87)} ∥ Q k − q k ∥ Y τ ∗ ≤ δ T , z ∈ K sup ∣ ρ k ( z ) − a k ( z ) ∣ ≤ δ B . (87) Each partition weight ρ k \rho_k ρ k K \mathcal{K} K [25 ] therefore gives a ReLU network a ~ k \widetilde{a}_k a k
To construct q ~ k \widetilde{q}_k q k I h Q k I_hQ_k I h Q k Theorem 4.1 , applied to each component of Q k Q_k Q k Q k Q_k Q k [ 0 , τ ∗ ] × T d [0,\tau^*]\times\mathbb{T}^d [ 0 , τ ∗ ] × T d
∥ Q k − I h Q k ∥ Y τ ∗ ≤ δ T 2 . \lVert Q_k-I_hQ_k\rVert_{Y_{\tau^*}}\leq\frac{\delta_T}{2}. ∥ Q k − I h Q k ∥ Y τ ∗ ≤ 2 δ T . The one-dimensional basis functions have exact ReLU representations. Approximating their products by Lemma 3.9 gives a vector-valued ReLU network q ~ k \widetilde{q}_k q k
∥ I h Q k − q ~ k ∥ Y τ ∗ ≤ δ T 2 . \lVert I_hQ_k-\widetilde{q}_k\rVert_{Y_{\tau^*}}\leq\frac{\delta_T}{2}. ∥ I h Q k − q k ∥ Y τ ∗ ≤ 2 δ T . The triangle inequality proves the first bound in (86). The network depends on the spatial variables through the periodic one-dimensional basis functions, so its values agree on opposite faces of [ 0 , 2 π ] d [0,2\pi]^d [ 0 , 2 π ] d q ~ k \widetilde{q}_k q k Y τ ∗ Y_{\tau^*} Y τ ∗
Define A ~ ( z ) : = ∑ k a ~ k ( z ) q ~ k \widetilde{A}(z):=\sum_k\widetilde{a}_k(z)\widetilde{q}_k A ( z ) := ∑ k a k ( z ) q k ∑ k ρ k ( z ) q ~ k \sum_k\rho_k(z)\widetilde{q}_k ∑ k ρ k ( z ) q k
∥ A ( z ) − A ~ ( z ) ∥ Y τ ∗ ≤ ∑ k ρ k ( z ) ∥ Q k − q ~ k ∥ Y τ ∗ + ∑ k ∣ ρ k ( z ) − a ~ k ( z ) ∣ ∥ q ~ k ∥ Y τ ∗ ≤ δ T + m ( B + δ T ) δ B . \begin{aligned}
\lVert A(z)-\widetilde{A}(z)\rVert_{Y_{\tau^*}}
&\leq\sum_k\rho_k(z)\lVert Q_k-\widetilde{q}_k\rVert_{Y_{\tau^*}}\\
&\quad+\sum_k\lvert\rho_k(z)-\widetilde{a}_k(z)\rvert\lVert\widetilde{q}_k\rVert_{Y_{\tau^*}}\\
&\leq\delta_T+m(B+\delta_T)\delta_B.
\end{aligned} ∥ A ( z ) − A ( z ) ∥ Y τ ∗ ≤ k ∑ ρ k ( z ) ∥ Q k − q k ∥ Y τ ∗ + k ∑ ∣ ρ k ( z ) − a k ( z )∣ ∥ q k ∥ Y τ ∗ ≤ δ T + m ( B + δ T ) δ B . The partition property bounds the first sum by δ T \delta_T δ T m m m m m m ( B + δ T ) δ B (B+\delta_T)\delta_B ( B + δ T ) δ B
δ T : = ε 8 , δ B : = ε 8 m ( B + 1 ) . \delta_T:=\frac{\varepsilon}{8},\qquad\delta_B:=\frac{\varepsilon}{8m(B+1)}. δ T := 8 ε , δ B := 8 m ( B + 1 ) ε . As ε < 1 \varepsilon<1 ε < 1 δ T + m ( B + δ T ) δ B ≤ ε / 4 \delta_T+m(B+\delta_T)\delta_B\leq\varepsilon/4 δ T + m ( B + δ T ) δ B ≤ ε /4 A ~ ( c J 0 x ( u 0 ) , c J ( ζ ) ) \widetilde{A}(c_{J_0}^{x}(u_0),c_J(\zeta)) A ( c J 0 x ( u 0 ) , c J ( ζ ))
With z = ( c J 0 x ( u 0 ) , c J ( ζ ) ) z=(c_{J_0}^{x}(u_0),c_J(\zeta)) z = ( c J 0 x ( u 0 ) , c J ( ζ ))
∥ S l o c ( u 0 , ζ ) − Γ θ N S E ( u 0 , ζ ) ∥ Y τ ∗ ≤ ∥ S l o c ( u 0 , ζ ) − S l o c ( u 0 , Π J ζ ) ∥ Y τ ∗ + ∥ S l o c ( u 0 , Π J ζ ) − S l o c ( Π J 0 x u 0 , Π J ζ ) ∥ Y τ ∗ + ∥ Γ J , J 0 N S E ( z ) − A ( z ) ∥ Y τ ∗ + ∥ A ( z ) − A ~ ( z ) ∥ Y τ ∗ ≤ ε 4 + ε 4 + ε 4 + ε 4 = ε . \begin{aligned}
\lVert\mathcal{S}^{\mathrm{loc}}(u_0,\zeta)-\Gamma_{\theta}^{\mathrm{NSE}}(u_0,\zeta)\rVert_{Y_{\tau^*}}
&\leq\lVert\mathcal{S}^{\mathrm{loc}}(u_0,\zeta)-\mathcal{S}^{\mathrm{loc}}(u_0,\Pi_J\zeta)\rVert_{Y_{\tau^*}}\\
&\quad+\lVert\mathcal{S}^{\mathrm{loc}}(u_0,\Pi_J\zeta)-\mathcal{S}^{\mathrm{loc}}(\Pi_{J_0}^{x}u_0,\Pi_J\zeta)\rVert_{Y_{\tau^*}}\\
&\quad+\lVert\Gamma_{J,J_0}^{\mathrm{NSE}}(z)-A(z)\rVert_{Y_{\tau^*}}+\lVert A(z)-\widetilde{A}(z)\rVert_{Y_{\tau^*}}\\
&\leq\frac{\varepsilon}{4}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}=\varepsilon.
\end{aligned} ∥ S loc ( u 0 , ζ ) − Γ θ NSE ( u 0 , ζ ) ∥ Y τ ∗ ≤ ∥ S loc ( u 0 , ζ ) − S loc ( u 0 , Π J ζ ) ∥ Y τ ∗ + ∥ S loc ( u 0 , Π J ζ ) − S loc ( Π J 0 x u 0 , Π J ζ ) ∥ Y τ ∗ + ∥ Γ J , J 0 NSE ( z ) − A ( z ) ∥ Y τ ∗ + ∥ A ( z ) − A ( z ) ∥ Y τ ∗ ≤ 4 ε + 4 ε + 4 ε + 4 ε = ε . Every bound is uniform over the initial-data class and the deterministic forcing ball in (80). This proves that estimate with one network for all these inputs.
Lastly, Fernique’s estimate in Lemma 2.3 , applied to the vector-valued Gaussian variable, gives
P ( G ρ c ) = P ( ∥ η ∥ X 0 > ρ / C ν ) ≤ C exp ( − c ρ 2 / C ν 2 ) ≤ C e − c ′ ρ 2 . \mathbb{P}(G_\rho^c)=\mathbb{P}(\lVert\eta\rVert_{X_0}>\rho/C_\nu)\leq C\exp(-c\rho^2/C_\nu^2)\leq Ce^{-c'\rho^2}. P ( G ρ c ) = P (∥ η ∥ X 0 > ρ / C ν ) ≤ C exp ( − c ρ 2 / C ν 2 ) ≤ C e − c ′ ρ 2 . Fix u 0 ∈ G 0 N S E u_0\in\mathcal{G}_0^{\mathrm{NSE}} u 0 ∈ G 0 NSE
Z u 0 : = { ∥ S l o c ( u 0 , η ) − Γ θ N S E ( u 0 , η ) ∥ Y τ ∗ , on G ρ , 0 , on G ρ c . Z_{u_0}:=
\begin{cases}
\lVert\mathcal{S}^{\mathrm{loc}}(u_0,\eta)-\Gamma_{\theta}^{\mathrm{NSE}}(u_0,\eta)\rVert_{Y_{\tau^*}}, & \text{on }G_\rho,\\
0, & \text{on }G_\rho^c.
\end{cases} Z u 0 := { ∥ S loc ( u 0 , η ) − Γ θ NSE ( u 0 , η ) ∥ Y τ ∗ , 0 , on G ρ , on G ρ c . By (73) , (74) , and the continuity of restriction from X 0 X_0 X 0 X 1 X_1 X 1 Z u 0 Z_{u_0} Z u 0 (81) gives Z u 0 ≤ ε 1 G ρ Z_{u_0} \le\varepsilon_1 G_\rho Z u 0 ≤ ε 1 G ρ
E Z u 0 ≤ ε P ( G ρ ) ≤ ε . \mathbb{E}Z_{u_0} \le\varepsilon\mathbb{P}(G_\rho) \le\varepsilon. E Z u 0 ≤ ε P ( G ρ ) ≤ ε . This bound is uniform in u 0 ∈ G 0 N S E u_0 \in\mathcal{G}_0^{\mathrm{NSE}} u 0 ∈ G 0 NSE (82) .
Department of Mathematics and Computer Science, Fisk University, Nashville, TN 37208, USA Email address: hdu@fisk.edu
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA Email address: qfeng2@fsu.edu
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA Email address: ww@ou.edu
