Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs)—and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of Escherichia coli bacterial motility. The fine scale, chemomechanical, hybrid (continuum—Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained “Keller–Segel class” chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be black-box (when no prior knowledge about the PDE law structure is assumed) or gray-box when parts of the equation (e.g. the pure diffusion part) is known and “hardwired” in the regression process. More importantly, we discuss data-driven corrections (both additive and functional), to analytically known, approximate closures.