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The data-driven discovery of partial differential equations (PDEs) consistentwith spatiotemporal data is experiencing a rebirth in machine learningresearch. Training deep neural networks to learn such data-driven partialdifferential operators requires extensive spatiotemporal data. For learningcoarse-scale PDEs from computational fine-scale simulation data, the trainingdata collection process can be prohibitively expensive. We propose totransformatively facilitate this training data collection process by linkingmachine learning (here, neural networks) with modern multiscale scientificcomputation (here, equation-free numerics). These equation-free techniquesoperate over sparse collections of small, appropriately coupled, space-timesubdomains ("patches"), parsimoniously producing the required macro-scaletraining data. Our illustrative example involves the discovery of effectivehomogenized equations in one and two dimensions, for problems with fine-scalematerial property variations. The approach holds promise towards making thediscovery of accurate, macro-scale effective materials PDE models possible byefficiently summarizing the physics embodied in "the best" fine-scalesimulation models available.

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