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Neural Network approximation power on homogeneous and heterogeneous reaction-diffusion equations

Haotian Feng

TL;DR

The paper provides a rigorous theoretical framework showing that neural networks can approximate reaction–diffusion PDE solutions with depth requirements that depend on dimensionality: a 2-layer network suffices in 1D, while a 3-layer network suffices in 2D, under Lipschitz/discretization assumptions. By discretizing the domain into subdomains and applying universal approximation results, the authors prove constructive approximation results for the Fisher–KPP equation in both homogeneous and heterogeneous media, and outline extensions to elliptic and parabolic PDEs. The key contributions are the precise NN-depth/size bounds tied to problem dimensionality, and the Lipschitz-based arguments that enable applying UAT to time-dependent RD problems via a time-independent elliptic reduction. This provides a theoretical foundation for NN-based PDE solvers and informs architectural choices for solving RD PDEs in practice.

Abstract

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.

Neural Network approximation power on homogeneous and heterogeneous reaction-diffusion equations

TL;DR

The paper provides a rigorous theoretical framework showing that neural networks can approximate reaction–diffusion PDE solutions with depth requirements that depend on dimensionality: a 2-layer network suffices in 1D, while a 3-layer network suffices in 2D, under Lipschitz/discretization assumptions. By discretizing the domain into subdomains and applying universal approximation results, the authors prove constructive approximation results for the Fisher–KPP equation in both homogeneous and heterogeneous media, and outline extensions to elliptic and parabolic PDEs. The key contributions are the precise NN-depth/size bounds tied to problem dimensionality, and the Lipschitz-based arguments that enable applying UAT to time-dependent RD problems via a time-independent elliptic reduction. This provides a theoretical foundation for NN-based PDE solvers and informs architectural choices for solving RD PDEs in practice.

Abstract

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.
Paper Structure (18 sections, 6 theorems, 19 equations, 2 figures)

This paper contains 18 sections, 6 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Problem Setup
  4. Domain discretization
  5. Neural Network Approximation Power Theorems
  6. 1D Universal Approximation Theorem
  7. High Dimensional Universal Approximation Theorem
  8. Reaction-Diffusion Equation
  9. Homogeneous Setting
  10. Heterogeneous Setting
  11. Assumptions
  12. Theoretical Analysis of Neural Network's approximation power
  13. Approximation of Fisher-KPP Equation
  14. Approximation on 1D Fisher-KPP Equation
  15. Approximation on high dimensional Fisher-KPP Equation
  16. ...and 3 more sections

Key Result

Lemma 1

For a continuous solution $u=f(x)$ in the domain of interest: when the domain is discretized into multiple connected subdomains, and the solution in each subdomain satisfies UTS, then the solution in whole domain also satistfies the UTS.

Figures (2)

  • Figure 1: Domain discretization of (a) 1D arbitrary curve (b) 2D arbitrary domain
  • Figure B.1: (a) 1-side ReLU function (b) 2-side ReLU function

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Lemma 4