Linking Machine Learning with Multiscale Numerics: DataDriven Discovery of Homogenized Equations
Abstract
The datadriven 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 datadriven partial differential operators requires extensive spatiotemporal data. For learning coarsescale PDEs from computational finescale 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, equationfree numerics). These equationfree techniques operate over sparse collections of small, appropriately coupled, spacetime subdomains ("patches"), parsimoniously producing the required macroscale training data. Our illustrative example involves the discovery of effective homogenized equations in one and two dimensions, for problems with finescale material property variations. The approach holds promise towards making the discovery of accurate, macroscale effective materials PDE models possible by efficiently summarizing the physics embodied in "the best" finescale simulation models available.
 Publication:

JOM  Journal of the Minerals, Metals and Materials Society
 Pub Date:
 December 2020
 DOI:
 10.1007/s11837020043998
 arXiv:
 arXiv:2008.11276
 Bibcode:
 2020JOM....72.4444A
 Keywords:

 Mathematics  Numerical Analysis;
 Physics  Computational Physics;
 35B27
 EPrint:
 JOM 2020