Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations
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https://doi.org/10.1007/s11837-020-04399-8 Ó 2020 The Minerals, Metals & Materials Society
AUGMENTING PHYSICS-BASED MODELS IN ICME WITH MACHINE LEARNING AND UNCERTAINTY QUANTIFICATION
Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations HASSAN ARBABI ,1,2 JUDITH E. BUNDER,3 GIOVANNI SAMAEY,4 ANTHONY J. ROBERTS,3 and IOANNIS G. KEVREKIDIS5,6 1.—Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, USA. 2.—Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, USA. 3.—School of Mathematical Sciences, University of Adelaide, Adelaide, Australia. 4.—NUMA, Department of Computer Science, University of Leuven (KU Leuven), Leuven, Belgium. 5.—Departments of Chemical and Biomolecular Engineering and Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, USA. 6.—e-mail: [email protected]
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.
INTRODUCTION Evolutionary models of materials dynamic behavior, in the form of Partial Differential Equations (PDEs), embodying conservation laws supplemented by appropriate closures, traditionally form the backbone of computational materials modeling. The requisite closures were initially mostly phenomenological, guided by experimentation, but in recent years such closures increasingly come from fine-scale, microscopic, possibly even ab initio computations. One of the signal promises of machine learning in its early days, and increasingly more so today, is the discovery of such evolutionary
(Received July 15, 2020; accepted September 21, 2020)
PDEs from spatiotemporal data—whether from physical observations or from computational observations obtained through fine-scale models. Yet materials problems are inherently multiscale: fine-scale models typically operate at the atomistic level; but the qu
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