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

2026-05-12

AI Summary

This work addresses the lack of theoretical foundations for neural operator approximation in nonlinear reaction-diffusion systems, such as the Gierer–Meinhardt model, by proposing a Laplacian spectral representation-based neural operator that learns the mapping from initial conditions to time-evolved solutions. Leveraging the spectral structure of the Green’s function, the study establishes the first explicit approximation error bounds for such systems and demonstrates that the required parameter complexity scales only polynomially with respect to the target accuracy, thereby mitigating the curse of dimensionality inherent in general operator learning. The theoretical analysis further reveals quantitative relationships among network depth, width, and the spectral rank of the solution. Numerical experiments confirm the superior accuracy and efficiency of the proposed Laplacian-based neural operator.

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.

Problem

• reaction-diffusion system
• neural operators
• approximation theory
• Gierer-Meinhardt model
• operator learning

Innovation

• Laplacian-based neural operators
• reaction-diffusion systems
• approximation error bounds
• spectral representation
• parameter complexity