Laplacian Eigenfunction-Based Neural Operator for Learning Nonlinear Reaction-Diffusion Dynamics

Abstract

Learning reaction-diffusion equations has become increasingly important across scientific and engineering disciplines, including fluid dynamics, materials science, and biological systems. In this work, we propose the Laplacian Eigenfunction-Based Neural Operator (LE-NO), a novel framework designed to efficiently learn nonlinear reaction terms in reaction-diffusion equations. LE-NO models the nonlinear operator on the right-hand side using a data-driven approach, with Laplacian eigenfunctions serving as the basis. This spectral representation enables efficient approximation of the nonlinear terms, reduces computational complexity through direct inversion of the Laplacian matrix, and alleviates challenges associated with limited data and large neural network architectures-issues commonly encountered in operator learning. We demonstrate that LE-NO generalizes well across varying boundary conditions and provides interpretable representations of learned dynamics. Numerical experiments in mathematical physics showcase the effectiveness of LE-NO in capturing complex nonlinear behavior, offering a powerful and robust tool for the discovery and prediction of reaction-diffusion dynamics.