Machine Learning for highly oscillatory differential equations

TL;DR

This paper introduces a machine learning approach to efficiently approximate vector fields in highly oscillatory differential equations, reducing pre-computation costs and combining neural networks with micro-macro techniques for improved solutions.

Contribution

It presents a novel method that uses neural networks to replace heavy pre-computations in solving oscillatory differential equations, enhancing efficiency.

Findings

Neural networks effectively approximate vector fields in oscillatory problems.

The combined approach improves computational efficiency over traditional methods.

Numerical simulations validate the method's accuracy and efficiency.

Abstract

Highly oscillatory differential equations, commonly encountered in multi-scale problems, are often too complex to solve analytically. However, several numerical methods have been developed to approximate their solutions. Although these methods have shown their efficiency, the first part of the strategy often involves heavy pre-computations from averaging theory. In this paper, we leverage neural networks (machine learning) to approximate the vector fields required by the pre-computations in the first part, and combine this with micro-macro techniques to efficiently solve the oscillatory problem. We illustrate our work by numerical simulations.