Neural Network approximation power on homogeneous and heterogeneous reaction-diffusion equations
It provides a theoretical foundation for neural network-based differential equation solvers, which is incremental as it builds on the universal approximation theorem.
This paper tackled the problem of insufficient theoretical understanding of why neural networks can approximate solutions to reaction-diffusion equations, demonstrating that two- and three-layer networks can approximate one- and two-dimensional versions, respectively.
Reaction-diffusion systems represent one of the most fundamental formulations used to describe a wide range of physical, chemical, and biological processes. With the increasing adoption of neural networks, recent research has focused on solving differential equations using machine learning techniques. However, the theoretical foundation explaining why neural networks can effectively approximate such solutions remains insufficiently explored. This paper provides a theoretical analysis of the approximation power of neural networks for one- and two-dimensional reaction-diffusion equations in both homogeneous and heterogeneous media. Building upon the universal approximation theorem, we demonstrate that a two-layer neural network can approximate the one-dimensional reaction-diffusion equation, while a three-layer neural network can approximate its two-dimensional counterpart. The theoretical framework presented here can be further extended to elliptic and parabolic equations. Overall, this work highlights the expressive power of neural networks in approximating solutions to reaction-diffusion equations and related PDEs, providing a theoretical foundation for neural network-based differential equation solvers.