Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations

TL;DR

This paper introduces a method that combines machine learning with multiscale numerics to efficiently discover homogenized PDEs from fine-scale simulation data, reducing data collection costs.

Contribution

It proposes linking neural networks with equation-free multiscale methods to generate macro-scale training data from limited fine-scale simulations.

Findings

Successfully discovered homogenized equations in 1D and 2D.

Reduced data collection costs for training neural networks.

Demonstrated effectiveness in capturing fine-scale physics at macro scale.

Abstract

The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential operators requires extensive spatiotemporal data. For learning coarse-scale PDEs from computational fine-scale simulation data, the training data collection process can be prohibitively expensive. We propose to transformatively facilitate this training data collection process by linking machine learning (here, neural networks) with modern multiscale scientific computation (here, equation-free numerics). These equation-free techniques operate over sparse collections of small, appropriately coupled, space-time subdomains ("patches"), parsimoniously producing the required macro-scale training data. Our illustrative example involves the discovery of effective homogenized equations in one and two dimensions, for problems with fine-scale material property variations. The approach holds promise towards making the discovery of accurate, macro-scale effective materials PDE models possible by efficiently summarizing the physics embodied in "the best" fine-scale simulation models available.