(346m) Learning Partial Differential Equations from Multiscale or Experimental Data: A Showcase on Bacterial Chemotaxis

E.coli chemotactic motion in the presence of a nutrient field has been extensively studied using wet-lab experiments [1], stochastic computational models [2] or partial differential equations [3]. The most challenging part in bridging these approaches, is the so-called chemotactic term, which describes how bacteria bias their motion up nutrient concentration gradients, as a result of a cascade of biochemical processes. Data-driven models can be used to learn either just the chemotactic term (gray box model), the entire right-hand-side of the chemotactic PDE (black box model) or the correction to a simplified constitutive model (closure model). This approach has been validated through computational data from established Monte Carlo simulations and continuum models. In this work, a machine-learning framework is proposed to uncover data-driven PDEs from multiscale or even experimental data. Even when the data at hand are sparse (coarse in space and/or time), noisy (inherent stochasticity in measurements) or partial (e.g. unaware of the spatiotemporal evolution of the nutrient field), we can attempt to learn the right hand side of a PDE for an evolving bacterial density. In fact, a second-order data-driven PDE in terms of the measurable bacterial density can encode the evolution of both the unmeasured nutrient field and the measured bacterial density (in the spirit of Whitney’s and Takens’s embedding theorems). The algorithmic pipeline proposed includes smoothing algorithms, interpolation with proper orthogonal decomposition/autoencoders and either a feedforward neural network or a Gaussian Process Regression learning of the dynamics. The resulting data-driven PDE can be integrated to predict/reproduce fine scale computational or experimental bacterial density profile data. Moreover, it can also be used to construct nonlinear observations of the (unmeasured) nutrient field evolution.

[1] T. Bhattacharjee, D. B. Amchin, J. A. Ott, F. Kratz, S. S. Datta, ”Chemotactic Migration of bacteria in Porous Media”, bioarxiv.org, 2020, doi: https://doi.org/10.1101/2020.08.10.244731

[2] S. Setayeshgar, C. W. Gear, H. G. Othmer, I. G. Kevrekidis, ”Application of Coarse Integration to Bacterial Chemotaxis”, Multiscale Model Simul., 2005, 4(1), 307-327.

[3] R. Erban, H. G. Othmer, ”From individual to collective behavior in bacterial chemotaxis”, SIAM J. Appl. Math., 2004, 65(2), 361-391.