A machine learning framework for data driven acceleration of computations of di erential equations

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 o ine training process by (approximately) minimizing suitable (possibly nonconvex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying di erential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational e ciency over standard numerical methods.