A machine learning framework for data driven acceleration of computations of differential equations

Abstract

We propose a machine learning framework to accelerate numerical computations oftime-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existingnumerical methods as artificial neural networks, with a set of trainable parameters. These parametersare 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 tobe always consistent with the underlying differential equation. Numerical experiments involving bothlinear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods.

Keywords

• machine learning| deep learning| differential equations| non-convex optimization|time-dependent problems