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

Abstract

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs) -and the closures that lead to themfrom 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.

Physics-agnostic and Physics-infused machine learning for thin films flows: modeling, and predictions from small data

“…Our work falls in the category of dynamical system identification [5][6][7][8]. Recently increased interest in PDE identification has led to the development of alternative algorithmic tools, such as sparse identification of nonlinear dynamical systems using dictionaries [9,10], PDE-net [11], physics-informed neural networks [12], and others [13][14][15] Our algorithmic approach can be implemented on data from detailed PDE simulations [16], agent-based modeling [17,18] or Lattice Boltzmann simulations [19,20] among others. Extensions of PDE identification including gray-box or closure identification (such as those explored in our work) have been studied in the context of various applications [16,17,[21][22][23][24][25][26].…”

“…Recently increased interest in PDE identification has led to the development of alternative algorithmic tools, such as sparse identification of nonlinear dynamical systems using dictionaries [9,10], PDE-net [11], physics-informed neural networks [12], and others [13][14][15] Our algorithmic approach can be implemented on data from detailed PDE simulations [16], agent-based modeling [17,18] or Lattice Boltzmann simulations [19,20] among others. Extensions of PDE identification including gray-box or closure identification (such as those explored in our work) have been studied in the context of various applications [16,17,[21][22][23][24][25][26]. In the relevant literature, the Kuramoto-Sivashinsky (KS) equation, selected in this work as a low-fidelity counterpart of the NS equations, has served as a benchmark case study, due to its wealth of dynamic responses and highly nonlinear nature [5,14,[27][28][29].…”