Neural networks for learning macroscopic chemotactic sensitivity from microscopic models

The macroscopic (population-level) dynamics of chemotactic cell movement -- arising from underlying microscopic (individual-based) models -- are often described by parabolic partial differential equations (PDEs) governing the spatio-temporal evolution of cell concentrations. In certain cases, these macroscopic PDEs can be analytically derived from microscopic models, thereby elucidating the dependence of PDE coefficients on the parameters of the underlying individual-based dynamics. However, such analytical derivations are not always feasible, particularly for more complex or nonlinear microscopic models. In these instances, neural networks offer a promising alternative for estimating the coefficients of macroscopic PDEs directly from data generated by microscopic simulations. In this work, three microscopic models of chemotaxis are investigated. The macroscopic chemotaxis sensitivity is estimated using neural networks, thereby bridging the gap between individual-level behaviours and population-level descriptions. The results are compared with macroscopic PDEs, which can be derived for each model in certain parameter regimes.