arXiv subjects

Maxime Bouchereau

Publications and source records attributed to Maxime Bouchereau.

2 recordsLinked to original sources

arXiv2026

Machine Learning for highly oscillatory differential equations

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.

math.NA↗

arXiv2023

Machine Learning Methods for Autonomous Ordinary Differential Equations

Ordinary Differential Equations are generally too complex to be solved analytically. Approximations thereof can be obtained by general purpose numerical methods. However, even though accurate schemes have been developed, they remain computationally expensive: In this paper, we resort to the theory of modified equations in order to obtain ''on the fly'' cheap numerical approximations. The recipe consists in approximating, prior to that, the modified field associated to the modified equation by neural networks. Elementary convergence results are then established and the efficiency of the technique is demonstrated on experiments.

math.NA↗