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

arXiv: https://arxiv.org/abs/2510.14094 · HTML: https://mlsanity.com/paper/2510.14094/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 16 references that are in the map.
- First posted on arXiv 15 Oct 2025.

## Builds on (most cited first, 11 of 11)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate](https://mlsanity.com/paper/2102.02617/) (2021)

## Abstract (arXiv)

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.