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

2018/07/25 by Siddhartha Mishra, Mishra, Siddhartha · 2 citations
Computer Science · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.1807.09519

openalex publication_date 2018/07/25 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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.

Citations

• An overview of gradient descent optimization algorithms

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