A Semigroup-Aware Kernel Neural Operator with Latent Exponential Integration for Forced PDEs

Abstract

We propose a semigroup-aware structured kernel neural operator for source-driven evolution PDEs. Unlike generic time-conditioned kernel models, the proposed formulation represents temporal dependence through a latent exponential propagator $e^{tA_θ}$, where $A_θ$ is a learned latent generator. This provides a structured transition from a non-ODE kernel representation to an ODE-inspired kernel operator whose latent temporal evolution satisfies the composition principle of evolution semigroups. For time-dependent forcing terms, a direct kernel-based evaluation requires repeated quadrature over delayed kernel evaluations. To address this computational bottleneck, we derive a latent-first reformulation of the source term: the forcing field is projected onto the kernel-induced latent space, where its accumulated effect is represented as the solution of a forced linear latent dynamical system. For a fixed learned kernel, this reformulation is exact. The resulting system can then be integrated using a $ϕ_1$-based exponential integrator, yielding a source-evaluation path that is structurally aligned with the learned latent propagator. The resulting framework combines geometry- and physics-conditioned low-rank kernel factorization with alternative source-evaluation paths based on either explicit Duhamel quadrature or latent exponential integration. Dissipativity constraints on the latent generator and identity-preserving corrections near the initial time are optional architectural components. Numerical experiments on advection-iffusion-reaction (ADR) problems compare the non-ODE and semigroup-aware formulations, evaluate the quadrature and $ϕ_1$ source paths, and assess predictive accuracy and computational cost.