Solving differential equations via artificial neural networks

Findings and failures in a model problem

authored by

Tobias Knoke, Thomas Wick

Abstract

In this work, we discuss some pitfalls when solving differential equations with neural networks. Due to the highly nonlinear cost functional, local minima might be approximated by which functions may be obtained, that do not solve the problem. The main reason for these failures is a sensitivity on initial guesses for the nonlinear iteration. We apply known algorithms and corresponding implementations, including code snippets, and present an example and counter example for the logistic differential equations. These findings are further substantiated with variations in collocation points and learning rates.

Organisation(s)

Institute of Applied Mathematics
PhoenixD: Photonics, Optics, and Engineering - Innovation Across Disciplines

External Organisation(s)

Université Paris-Saclay

Type

Article

Journal

Examples and Counterexamples

Volume

1

Publication date

11.2021

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Computational Mathematics, Applied Mathematics, Mathematics (miscellaneous)

Electronic version(s)

https://doi.org/10.1016/j.exco.2021.100035 (Access: Open)