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

TL;DR

This work tackles how infection status and environmental resources shape dispersal in insect populations by learning a data-driven mean-field PDE for population density. It extends WSINDy with kernel density estimation to infer a McKean–Vlasov-type Fokker–Planck equation from sparse 2D-larval position data, yielding $u_t = \nabla \cdot (u(\nabla \mathcal{V} + \nabla \mathcal{K} * u) + \mathbf{D} \nabla u)$. The results show diffusion-dominated dynamics with the external environmental potential $\mathcal{V}$ capturing plant spacing, while the nonlinear interaction $\mathcal{K}$ is weaker, and ensemble data can justify richer McKean–Vlasov models via ${\Delta}{\rm AIC}$ improvements. This framework provides quantitative dispersal metrics (e.g., diffusion tensor $\mathbf{D}$ and effective diffusion) across infection and resource conditions, offering actionable insights for outbreak forecasting and pest-management strategies.

Abstract

Insect species subject to infection, predation, and anisotropic environmental conditions may exhibit preferential movement patterns. Given the innate stochasticity of exogenous factors driving these patterns over short timescales, individual insect trajectories typically obey overdamped stochastic dynamics. In practice, data-driven modeling approaches designed to learn the underlying Fokker-Planck equations from observed insect distributions serve as ideal tools for understanding and predicting such behavior. Understanding dispersal dynamics of crop and silvicultural pests can lead to a better forecasting of outbreak intensity and location, which can result in better pest management. In this work, we extend weak-form equation learning techniques, coupled with kernel density estimation, to learn effective models for lepidopteran larval population movement from highly sparse experimental data. Galerkin methods such as the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm have recently proven useful for learning governing equations in several scientific contexts. We demonstrate the utility of the method on a sparse dataset of position measurements of fall armyworms (Spodoptera frugiperda) obtained in simulated agricultural conditions with varied plant resources and infection status.

Figures (16)

• Figure 1: (Left) Illustrating the forces at play in eq. (\ref{['eq:SDE']}). (Right) A fall armyworm larva.
• Figure 1: Visualizing the learned environmental potential $\mathcal{V}$ and interaction potential $\mathcal{K}$ for the empirical distributions $\mu(\boldsymbol{x}; \mathbf{X}_t)$ and $\mu(\boldsymbol{x}; \mathbf{X}_t^{3,1})$, respectively. Note that the learned $\mathcal{V}$ resembles the soybean plant spacing in each domain.
• Figure 1: A hyperparameter sweep illustrating the sensitivity of the effective diffusion constants $D_{\text{eff}}$ predicted by WSINDy to changes in the test function support radii $\boldsymbol{m} = (m_x, m_y, m_t)$. Here, we use $m_x = m_y$ and plot an '$\boldsymbol{\times}$' at $(m_x,m_t) = (10,6)$.
• Figure 2: Illustrating the dynamics that are possible under an SDE consistent with eq. (\ref{['eq:simplified_diffusion_focused_pde']}) in various parameter regimes. Here, we fix the diffusion strength $\Pi_D = 1$ and incrementally increase the potential strengths $\Pi_V$ and $\Pi_K$; see §\ref{['sec:additional_implementation_details']}.
• Figure 2: Illustrating the weak-form feature $(\psi * u)(\boldsymbol{x},t)$ at time $t=1$ for the ensemble distribution $\mu(\mathbf{X}_t)$, given the chosen test function support radii $\boldsymbol{m}=(10, 10, 6)$. We manually select test function spectra $|\hat{\psi}|$ that induce minimal smoothing.