Mathematical modelling of biological population dynamics often involves
proposing high fidelity discrete agent-based models that capture stochasticity
and individual-level processes. These models are often considered in
conjunction with an approximate coarse-grained differential equation that
captures population-level features only. These coarse-grained models are only
accurate in certain asymptotic parameter regimes, such as enforcing that the
time scale of individual motility far exceeds the time scale of birth/death
processes. When these coarse-grained models are accurate, the discrete model
still abides by conservation laws at the microscopic level, which implies that
there is some macroscopic conservation law that can describe the macroscopic
dynamics. In this work, we introduce an equation learning framework to find
accurate coarse-grained models when standard continuum limit approaches are
inaccurate. We demonstrate our approach using a discrete mechanical model of
epithelial tissues, considering a series of four case studies that illustrate
how we can learn macroscopic equations describing mechanical relaxation, cell
proliferation, and the equation governing the dynamics of the free boundary of
the tissue. While our presentation focuses on this biological application, our
approach is more broadly applicable across a range of scenarios where discrete
models are approximated by approximate continuum-limit descriptions. All code
and data to reproduce this work are available at
https://github.com/DanielVandH/StepwiseEQL.jl.Comment: 42 pages, 18 figure