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

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 c...

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Bibliographic Details
Published in:JOM (1989) Vol. 72; no. 12; pp. 4444 - 4457
Main Authors: Arbabi, Hassan, Bunder, Judith E., Samaey, Giovanni, Roberts, Anthony J., Kevrekidis, Ioannis G.
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2020
Springer Nature B.V
Subjects:
Applied mathematics
Artificial neural networks
Augmenting Physics-based Models in ICME with Machine Learning and Uncertainty Quantification
Chemistry/Food Science
Data collection
Earth Sciences
Engineering
Environment
Homogenization
Machine learning
Material properties
Mathematical models
Neural networks
Operators (mathematics)
Partial differential equations
Physics
Scale models
Simulation
Spacetime
Training
ISSN:1047-4838, 1543-1851
Online Access:Get full text
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Summary: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.
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ISSN:1047-4838
1543-1851
DOI:10.1007/s11837-020-04399-8