A generalization error bound of physics-informed neural networks for ecological diffusion models

Abstract

Ecological diffusion equations (EDEs) are partial differential equations (PDEs) that model spatiotemporal dynamics, often applied to wildlife diseases. Derived from ecological mechanisms, EDEs are useful for forecasting, inference, and decision-making, such as guiding surveillance strategies for wildlife diseases. Physics-informed neural networks (PINNs) offer a cutting-edge approach to approximating the solution of EDEs by constraining the loss function. PINNs have shown empirical success in forecasting and inference in wildlife disease problems. However, theoretical properties of PINNs in this context remain unexplored. In this paper, we derive a generalization error bound for PINNs applied to forward problems in EDEs, providing new crucial theoretical results for statistical analysis. Furthermore, we compare the performance of PINNs with commonly used numerical solvers, showing that PINNs achieve competitive accuracy while providing important modelling flexibility. Our results help connecting theory and application in EDE-based statistical models for ecological systems.