(340e) Learning Coarse-Grained Partial Differential Equations from Fine-Scale Data Via Machine Learning

For many fine-grained (atomistic, stochastic, agent-based) systems, it is often desired to obtain reduced, coarse-grained, macroscopic governing equations (PDEs or ODEs) that can accelerate computation at the system level. Yet, analytic extraction of coarse-scale PDEs or ODEs from fine-scale observations is typically possible only for a limited class of systems. On the other hand, the continuing growth of computational and observational resources has enabled a massive collection of data from many fine-scale systems, both in experiments and in computations. In this talk, we present a framework to identify corresponding coarse-scale governing equations (e.g. Partial Differential Equations or PDEs) purely from data. In the first step, we extract salient coarse-scale observables via feature selection methods and manifold learning techniques (such as Diffusion Maps). Then we use several machine learning algorithms (Gaussian processes, Artificial Neural Networks, and/or Diffusion Maps) to discover the underlying coarse-scale PDEs from those observables [1].

We demonstrate the application of our framework in particle-based models of fluid flow transport and of bacterial chemotaxis phenomena. We identify a set of coarse-scale PDEs from fine-scale data obtained either from particle/agent-based models, or possibly also from experiments. We compare the learned PDEs to established models (like the viscous Burgers or the Keller-Segel (KS) PDEs [2] respectively). Our numerical experiments show that our data-driven framework can model the flow/chemotaxis phenomena qualitatively (and quantitatively) comparable to the known macroscopic PDEs. A special feature of our framework is that it can provide multiple PDE forms at the macroscopic level that are ``equally good" at describing the data. For example, contrary to the KS PDEs that treat bacteria and chemoattractant density as independent variables, we can formulate alternative PDEs in the form of a single density variable, its spatial derivatives, and its time-delayed history. This formulation allows flexibility in computation and control depending on the availability of measurements.

[1] Lee, S., Kooshkbaghi, M., Spiliotis, K., Siettos, C.I. and Kevrekidis, I.G. (2020), Coarse-scale PDEs from fine-scale observations via machine learning, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(1), p.013141.

[2] Keller, E.F. and Segel, L.A. (1970), Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26(3), pp.399-415.