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

TL;DR

This paper analyzes neural operators for reaction-diffusion PDEs, providing explicit approximation bounds and demonstrating their efficiency in modeling complex nonlinear systems.

Contribution

It establishes explicit error bounds for Laplacian-based neural operators applied to reaction-diffusion systems, highlighting their polynomial complexity growth.

Findings

Parameter complexity grows polynomially with accuracy

Laplacian spectral representation improves approximation efficiency

Numerical experiments support theoretical bounds

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.