Using Machine Learning to Design Time Step Size Controllers for Stable Time Integrators

TL;DR

This paper introduces a machine learning-based approach using Bayesian optimization to design effective time step controllers for stable time integrators, improving stability and efficiency in solving differential equations.

Contribution

It proposes a novel method combining Bayesian optimization with error estimation for designing time step controllers, applicable to various stable integrators.

Findings

Controllers outperform classical PI and PID controllers.

Optimized controllers achieve comparable or better stability and efficiency.

Method effective for both ODEs and PDEs.

Abstract

We present a new method for developing time step controllers based on a technique from the field of machine learning. This method is applicable to stable time integrators that have an embedded scheme, i.e., that have local error estimation similar to Runge-Kutta pairs. To design good time step size controllers using these error estimates, we propose to use Bayesian optimization. In particular, we design a novel objective function that captures important properties such as tolerance convergence and computational stability. We apply our new approach to several modified Patankar--Runge--Kutta (MPRK) schemes and a Rosenbrock-type scheme, equipping them with controllers based on digital signal processing which extend classical PI and PID controllers. We demonstrate that the optimization process yields controllers that are at least as good as the best controllers chosen from a wide range of suggestions available for classical explicit and implicit time integration methods by providing work-precision diagrams on a variety of ordinary and partial differential equations.