Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
This work develops a rigorous theory for neural operators applied to nonlinear reaction-diffusion systems, specifically using Laplacian eigenfunction bases. The method provides explicit error bounds showing that network depth grows logarithmically and spectral rank grows polynomially. This demonstrates that this architecture effectively exploits the PDE structure to avoid complexity issues common in other learning methods.