Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data
This work addresses the challenge of deriving PDEs from stochastic simulations for bacterial chemotaxis, which is incremental as it applies existing ML methods to a specific domain problem.
The authors tackled the problem of discovering macroscopic chemotactic PDEs from agent-based simulations of E.coli motility, using a machine learning framework with Automatic Relevance Determination and regressors like neural networks and Gaussian Processes to learn black- or gray-box equations, achieving effective coarse-grained models.
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 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 individual cells. We exploit Automatic Relevance Determination (ARD) within a Gaussian Process framework for the identification of a parsimonious set of collective observables that parametrize the law of the effective PDEs. Using these observables, in a second step 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. We also discuss data-driven corrections (both additive and functional) of analytically known, approximate closures.