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

Abstract

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.

References in corpus (3)

• An overview of gradient descent optimization algorithms
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• An Efficient Deep Learning Technique for the Navier-Stokes Equations: Application to Unsteady Wake Flow Dynamics

Cited by in corpus (4)

• Deep learning observables in computational fluid dynamics
• Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks
• An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradient
• Modelling parametric uncertainty in PDEs models via Physics-Informed Neural Networks