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

2025-10-09

AI Summary

Traditional modeling approaches for pest dispersal dynamics—e.g., in agricultural and forest systems (such as *Spodoptera frugiperda*)—fail under sparse experimental trajectory data. Method: We propose a novel framework integrating weak-form equation learning with adaptive kernel density estimation, enabling direct inference of nonlinear Fokker–Planck-type mean-field PDEs from highly sparse individual-level trajectories. Our approach combines weak SINDy (WSINDy) for sparse model identification, Galerkin projection for weak-form enforcement, and adaptive kernel density estimation to reconstruct spatiotemporal evolution of population probability distributions. Contribution/Results: Validated in simulated agroecosystems, the model accurately captures collective movement under infection, predation, and anisotropic environmental influences. It significantly improves predictive accuracy for both outbreak intensity and spatial location of pest aggregations. This work establishes an interpretable, generalizable, data-driven dynamical modeling paradigm for precision pest management.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Relational Probabilistic Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Learning Fokker-Planck equations from sparse insect movement data
• Modeling preferential insect dispersal under environmental influences
• Forecasting pest outbreak dynamics using weak-form equation learning

Innovation

• Weak-form equation learning for mean-field PDEs
• Kernel density estimation for sparse data
• WSINDy algorithm for insect movement modeling