Learning Structured Population Models from Data with WSINDy

In the context 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 method uses 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 process of the model directly from the data. We additionally provide a cross-validation method which helps fine tune the recovered boundary process to the data. Several test cases are considered, demonstrating the method's performance for different previously studied models, 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.