On the Data-Driven Discovery and Calibration of Closures

We explore the data-driven learning of closures in problems involving bacterial chemotactic motion. The data come from agent-based stochastic computations as well as from experiments, and are validated through computational data from established continuum models. Machine learning has long been used to learn unknown PDEs from spatiotemporal data [1] and recently there has been a renewed interest, especially for the case of multiscale simulation data [2]. Starting from neural network architectures that learn a PDE from spatiotemporal data as a "black box", we proceed to learn only parts of this evolution law as a "gray box", and, in particular, the unknown closure terms.

Using agent-based simulation data and experimental data from E. Coli bacterial chemotaxis, we explore the data-driven discovery of the chemotactic terms; find different equivalent parametrizations of them, and also show how to calibrate approximate (qualitatively correct but quantitatively inaccurate) closures to "the truth" in a data-driven manner.

We also show how the procedure can be linked with equation-free multiscale computational techniques (like patch dynamics) to help collect the necessary macroscale data in the most parsimonious way possible. Possibly the most important finding is that several different "equivalent on the data" PDEs as well as several different "equivalent on the data" closures can be identified that are consistent with the observations. We will also discuss the explainability (in terms of humanly understandable terms) of the variables and parametrizations of the discovered closures. The work forms a bridge between analytical/mechanistic understanding, and data-driven "black box" learning of physical process dynamics, allowing for a synergy between the two options.

[1] Gonzalez-Garcia, R., Rico-Martinez, R. and Kevrekidis, I.G., (1998). Identification of distributed parameter systems: A neural net based approach. Computers & chemical engineering, 22, pp.S965-S968

[2] 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.