Accelerating Explicit Time-Stepping with Spatially Variable Time Steps Through Machine Learning

Use of machine learning (ML) to solve partial differential equations (PDEs) is a growing area of research. In this work we apply ML to accelerate a numerical discretization scheme: the radial basis function time domain (RBF-TD) method. The RBF-TD method uses scattered nodes in both space and time to time-step PDEs. Here we investigate replacing a costly L1 minimization step in the original RBF-TD method with an ensemble ML model known as an extremely randomized trees regressor (ERT). We show that an ERT model trained on a simple PDE can maintain the high order accuracy of the original method with often significant speed up, while generalizing to a variety of convection-dominated PDEs. Through this work, we also extend the RBF-TD method to problems in 2-D space plus time. This study illustrates a novel opportunity to use ML to augment finite difference-related approximations while maintaining high order convergence under node refinement.