ArXiv

Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System

2026-05-12 · 12:13 UTC ·Takashi Furuya, Ryo Ozawa, Jenn-Nan Wang ·1 min read

Authors: Takashi Furuya, Ryo Ozawa, Jenn-Nan Wang
Categories: cs.LG, stat.ML
arXiv: https://arxiv.org/abs/2605.12025v1
PDF: https://arxiv.org/pdf/2605.12025v1

Brief

Laplacian-based neural operators are analyzed for the generalized Gierer–Meinhardt reaction–diffusion system: the paper obtains explicit approximation-error bounds depending on network depth, width, and spectral rank by using the Laplacian eigenfunction expansion of the PDE Green’s function. The authors show parameter complexity scales at most polynomially with accuracy and present numerical experiments consistent with theory. Summary based on the abstract; full text not reviewed.

Why it matters

The authors derive explicit approximation error bounds for the solution operator mapping initial conditions to time-dependent solutions of a generalized Gierer–Meinhardt reaction–diffusion system, expressed in terms of network depth, width, and spectral rank.

Key details

By exploiting the Laplacian eigenfunction (spectral) representation of the PDE Green's function, the paper proves required parameter complexity grows at most polynomially with target accuracy (alleviating a curse of parametric complexity) and reports numerical experiments that support the theoretical bounds. (Authors: Takashi Furuya, Ryo Ozawa, Jenn-Nan Wang; arXiv:2605.12025v1; published 2026-05-12.)

Source evidence

Abstract

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.