Particles to Partial Differential Equations Parsimoniously

TL;DR

This paper introduces a neural network and multiscale computation framework to efficiently discover macro-scale PDEs from microscopic simulations, reducing data requirements and identifying suitable variables through manifold learning and optimal transport.

Contribution

It combines equation-free numerics with data-driven methods to discover effective PDEs at macro scales directly from microscopic data, with reduced computational effort.

Findings

Successfully extracted coarse-grained PDEs from particle simulations.

Reduced data collection needs through sparse sampling.

Identified macro-scale variables using manifold learning and optimal transport.

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.