Enhancing Neural Network Differential Equation Solvers

2022/12/28 by Matthew J. H. Wright, Wright, Matthew J. H. · 1 citation
Engineering · Physics and Astronomy · #Advanced Numerical Analysis Techniques #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2301.13146

openalex publication_date 2022/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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