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.

本文提出了一种机器学习框架来从高精度随机模拟中学习大尺度化学趋化偏微分方程及其闭合问题，使用自适应相关决策根据高维数据选择合适的共同特征来简化该问题并使用机器学习的方式学习宏观大尺度的效应。