We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods.
@article{arxiv.1807.09519,
title = {A machine learning framework for data driven acceleration of computations of differential equations},
author = {Siddhartha Mishra},
journal= {arXiv preprint arXiv:1807.09519},
year = {2019}
}