Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
Takashi Furuya, Ryo Ozawa, Jenn-Nan Wang
TLDR
This paper shows Laplacian-based neural operators efficiently approximate reaction-diffusion systems with polynomial complexity.
Key contributions
- Analyzes Laplacian-based neural operators for Gierer-Meinhardt reaction-diffusion systems.
- Derives explicit error bounds based on network depth, width, and spectral rank.
- Exploits Laplacian spectral representation of the PDE's Green's function.
- Shows polynomial parameter complexity, alleviating the curse of dimensionality in operator learning.
Why it matters
This paper fills a gap in approximation theory for nonlinear reaction-diffusion systems using neural operators. It demonstrates how Laplacian-based architectures achieve efficient learning with polynomial complexity, addressing a key challenge in operator learning.
Original 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.
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