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We propose a machine learning framework to accelerate numerical computationsof 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 anoffline training process by (approximately) minimizing suitable (possiblynon-convex) loss functions by (stochastic) gradient descent methods. Theproposed algorithm is designed to be always consistent with the underlyingdifferential equation. Numerical experiments involving both linear andnon-linear ODE and PDE model problems demonstrate a significant gain incomputational efficiency over standard numerical methods.

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