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

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.

References in corpus (9)

• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs
• Learning data driven discretizations for partial differential equations
• iMapD: intrinsic Map Dynamics exploration for uncharted effective free energy landscapes
• Variational system identification of the partial differential equations governing pattern-forming physics: Inference under varying fidelity and noise
• Coarse-scale PDEs from fine-scale observations via machine learning
• Variable-free exploration of stochastic models: a gene regulatory network example
• An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning

Cited by in corpus (9)

• A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations
• Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems
• Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference
• Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs
• Artificial Intelligence in Materials Science and Engineering: Current Landscape, Key Challenges, and Future Trajectorie
• Machine learning based data-driven discovery of nonlinear phase-field dynamics
• Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations
• Global and Local Reduced Models for Interacting, Heterogeneous Agents
• Learning the Latent dynamics of Fluid flows from High-Fidelity Numerical Simulations using Parsimonious Diffusion Maps