Learning structured population models from data with WSINDy

Characteristics of individuals in a population, such as age and size, play a key role in determining how populations change over time. In contexts of population dynamics, identifying effective model features, such as fecundity and mortality rates, is generally a complex and computationally intensive process, especially when the dynamics are heterogeneous across the population. In this work, we propose a Weak form Scientific Machine Learning-based method for selecting appropriate model ingredients from a library of scientifically feasible functions used to model structured populations. This paper presents extensions of the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) method to select the best-fitting ingredients from noisy time-series histogram data. This extension includes learning heterogeneous dynamics and also learning the boundary processes (such as birth) of the model directly from the data. We additionally incorporate a cross-validation method which helps fine tune the recovered boundary process hyperparameters to the data. Several test cases are considered, demonstrating the method's performance for several standard models from population modeling, including age and size-structured models. Through these examples, we examine both the advantages and limitations of the method, with a particular focus on the distinguishability of terms in the library.