Particles to Partial Differential Equations Parsimoniously

Abstract

Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at much coarser, meso- or macroscopic length scales. Discovering those coarse-grained effective PDEs can lead to considerable savings in computation-intensive tasks like prediction or control. We propose a framework combining artificial neural networks with multiscale computation, in the form of equation-free numerics, for efficient discovery of such macro-scale PDEs directly from microscopic simulations. Gathering sufficient microscopic data for training neural networks can be computationally prohibitive; equation-free numerics enable a more parsimonious collection of training data by only operating in a sparse subset of the space-time domain. We also propose using a data-driven approach, based on manifold learning and unnormalized optimal transport of distributions, to identify macro-scale dependent variable(s) suitable for the data-driven discovery of said PDEs. This approach can corroborate physically motivated candidate variables, or introduce new data-driven variables, in terms of which the coarse-grained effective PDE can be formulated. We illustrate our approach by extracting coarse-grained evolution equations from particle-based simulations with a priori unknown macro-scale variable(s), while significantly reducing the requisite data collection computational effort.

References in corpus (11)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data
• Learning data driven discretizations for partial differential equations
• Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
• The Gap-Tooth Method in Particle Simulations
• Deep learning to discover and predict dynamics on an inertial manifold
• Coarse-scale PDEs from fine-scale observations via machine learning
• Unnormalized Optimal Transport
• Variable-free exploration of stochastic models: a gene regulatory network example
• An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning
• Learning emergent PDEs in a learned emergent space

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