Efficient time stepping for numerical integration using reinforcement learning

TL;DR

This paper introduces a data-driven time-stepping controller that enhances classical numerical schemes for differential equations, improving efficiency and generalization over traditional adaptive methods, especially for complex systems.

Contribution

It combines classical quadrature rules with machine learning controllers to tailor numerical integration schemes to specific problem classes, outperforming existing adaptive methods.

Findings

Superior efficiency demonstrated in multiple examples

Better generalization to unseen initial data

Outperforms state-of-the-art adaptive schemes

Abstract

Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it either results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge--Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave sub-optimal when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalises better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.