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 etAθ, where Aθ 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 ϕ1-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 ϕ1 source paths, and assess predictive accuracy and computational cost.
Related
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BibTeX
@misc{abbas2026semigroupaware,
title = {{A Semigroup-Aware Kernel Neural Operator with Latent Exponential Integration for Forced PDEs}},
author = {Naseem Abbas and Vittorio Colao and Davide Macrì and Matthieu Barreau and William Spataro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39845},
eprint = {2610.39845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39845},
}
RIS
TY - UNPB
TI - A Semigroup-Aware Kernel Neural Operator with Latent Exponential Integration for Forced PDEs
AU - Naseem Abbas
AU - Vittorio Colao
AU - Davide Macrì
AU - Matthieu Barreau
AU - William Spataro
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39845
UR - https://arxiv.org/abs/2610.39845
AB - 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.
ER -