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We propose a machine learning framework for the data-driven discovery ofmacroscopic chemotactic Partial Differential Equations (PDEs) -- and theclosures that lead to them -- from high-fidelity, individual-based stochasticsimulations of E.coli bacterial motility. The fine scale, detailed, hybrid(continuum - Monte Carlo) simulation model embodies the underlying biophysics,and its parameters are informed from experimental observations of individualcells. We exploit Automatic Relevance Determination (ARD) within a GaussianProcess framework for the identification of a parsimonious set of collectiveobservables that parametrize the law of the effective PDEs. Using theseobservables, in a second step we learn effective, coarse-grained "Keller-Segelclass" chemotactic PDEs using machine learning regressors: (a) (shallow)feedforward neural networks and (b) Gaussian Processes. The learned laws can beblack-box (when no prior knowledge about the PDE law structure is assumed) orgray-box when parts of the equation (e.g. the pure diffusion part) is known and"hardwired" in the regression process. We also discuss data-driven corrections(both additive and functional) of analytically known, approximate closures.

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