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.

Keywords

modified equation, ordinary differential equation, neural network, numerical method, convergence analysis

2010 Mathematics Subject Classification

65L05, 65L70, 68Txx

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Received 14 April 2023

Received revised 12 October 2023

Accepted 21 December 2023

Published 18 July 2024