Enhancing Neural Network Differential Equation Solvers

TL;DR

This paper demonstrates how neural networks can be used to approximate solutions to differential equations, providing theoretical guarantees and practical strategies to improve accuracy, validated through numerical experiments on Poisson's equation.

Contribution

It introduces a theoretical framework for neural network-based differential equation solvers with error correction strategies and validates them with numerical experiments.

Findings

Neural networks can approximate solutions arbitrarily close to exact solutions.

Error correction networks significantly improve solution accuracy.

Numerical experiments confirm the effectiveness of proposed strategies.

Abstract

We motivate the use of neural networks for the construction of numerical solutions to differential equations. We prove that there exists a feed-forward neural network that can arbitrarily minimise an objective function that is zero at the solution of Poisson's equation, allowing us to guarantee that neural network solution estimates can get arbitrarily close to the exact solutions. We also show how these estimates can be appreciably enhanced through various strategies, in particular through the construction of error correction networks, for which we propose a general method. We conclude by providing numerical experiments that attest to the validity of all such strategies for variants of Poisson's equation.