Optimal Parameter Upscaling for Partial Differential Equation Models in Mathematical Biology

Abstract

Partial differential equation models in mathematical biology often involve space-dependent parameters, such as diffusion coefficients and advection fields, that cannot be measured explicitly and are therefore uncertain. In this work, we compute spatially adaptive, lower-dimensional approximations of these fields, using machine learning tools. Such parsimonious representations of the parameter space would greatly improve the efficiency of the resulting stochastic simulations, allow for more targeted use of reduced order models, and aid in the related design of interventions. Numerical examples demonstrate our theoretical results.