Efficient time stepping for numerical integration using reinforcement learning

Consequently, schemes for numerical discretization are a key element of scientific computing, and one of the main challenges is to determine a good trade-off between the required accuracy and numerical efficiency. While the standard approach to developing quadrature rules and numerical schemes to solve ODEs is based on Taylor series expansions and the associated error bounds determined by higher-order derivatives [2], the advances in data science and machine learning have recently fueled the development of alternative concepts that are based on training data. Most of these approaches are developed with the aim to efficiently compute the numerical solution of dynamical systems of high complexity, see, for instance, [18, 25, 15, 19, 20, 12], where the flow map F that takes a state x at time t to a future state x(t + t) is approximated. In contrast to that, our work addresses the task of efficiently performing numerical integration for integrands or differential equations of a given problem class to a desired accuracy. To this end, the next sample point at which the integrand or force term is evaluated is determined by finding an optimal trade-off between the two conflicting criteria accuracy and numerical efficiency. This task is carried out by a reinforcement learning algorithm which, taking past function evaluations and learned knowledge about the problem class into account, determines the next sample point at which to evaluate the function of interest. The efficiency of the proposed approach in comparison to state-of-the-art methods will be demonstrated using examples from the area of computing integrals (quadrature) as well as numerically solving differential equations.