A machine learning framework for data driven acceleration of computations of differential equations
This work addresses computational bottlenecks in solving differential equations, which is incremental as it builds on existing numerical methods.
The authors tackled the problem of accelerating numerical computations for time-dependent ODEs and PDEs by proposing a machine learning framework that recasts existing numerical methods as neural networks, resulting in a significant gain in computational efficiency over standard methods.
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.