Weak-Form Learning for Mean-Field Partial Differential Equations: An Application to Insect Movement

Abstract

Abstract. Insect populations subject to epidemic infection, predation, and anisotropic environmental conditions may exhibit preferential movement patterns. Given the virtually random nature of the exogenous factors driving these patterns over short timescales, individual insect trajectories often resemble overdamped stochastic processes. Consequently, data-driven modeling approaches designed to learn effective mean-field governing equations from observed insect populations serve as ideal tools for understanding and predicting such behavior. In this work, we extend existing weak-form equation learning techniques to learn effective partial differential equation models for lepidopteran larval population movement directly from sparse experimental data. We demonstrate the utility of the method on an experimental dataset obtained in simulated agricultural conditions. Relevance to Life Sciences. Understanding dispersal dynamics of crop and silvicultural pests can lead to improved forecasting of outbreak intensity and location, which results in better pest management. As a case study, we demonstrate the utility of our modeling methodology on a sparse experimental dataset consisting of position measurements of fall armyworms ( Spodoptera frugiperda). The data were obtained in simulated agricultural conditions with varied resource quality and larval infection status with respect to a natural pathogen. Using our weak-form modeling methodology, we characterize the dominant mechanisms in the dispersal dynamics and provide quantitative estimates of the effective diffusion rates. Our results indicate that the dispersal dynamics are primary diffusive, although nonnegligible contributions arise from a nonuniform plant resource distribution. Mathematical Content. Galerkin equation learning methods, such as the Weak-form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm, have recently proven useful for identifying mean-field governing equations from interacting particle data within several biological contexts. In this work, we adapt the WSINDy algorithm, coupled with kernel density estimation, to learn effective partial differential equation models for lepidopteran larval population movement directly from sparse experimental data. In particular, our method characterizes effective external and interaction potentials, as well as diffusive terms, in nonlinear Fokker–Planck models arising from a system of McKean–Vlasov stochastic ordinary differential equations.