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

TL;DR

This paper introduces a machine learning framework that transforms traditional numerical methods for solving differential equations into trainable neural networks, significantly speeding up computations while maintaining accuracy.

Contribution

It presents a novel approach to accelerate differential equation solvers by recasting them as neural networks trained offline, ensuring consistency with the original equations.

Findings

Significant computational speed-up over standard methods.

Applicable to both linear and nonlinear ODEs and PDEs.

Maintains consistency with the underlying 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.