Mechanism-AI Coupled Dynamic Systems in Mathematical Biology: Learning, Explanation, Reliability, and Applications

Source abstract

Mechanistic dynamic models are central to mathematical biology, but their practical use is often limited by unknown biological mechanisms, partially observed states, sparse and noisy data, and uncertainty in learned parameters. Deep learning provides flexible universal approximators, while mechanistic models provide structure, interpretability, and scientific constraints. This paper reviews and organizes a line of work on mechanism-AI coupled dynamic systems, with emphasis on foundations and applications in epidemic modeling. We discuss learning methods based on universal differential equations and physics-informed neural networks, explanation methods based on symbolic regression and sparse model discovery, reliability analysis through convergence and uncertainty quantification, and representative applications including effective reproduction number estimation, behavioral-change mechanism discovery, optimal epidemic control, last-layer Bayesian uncertainty quantification, and physics-informed reconstruction of threshold quantities. The goal is not to replace mathematical models by purely empirical predictors, but to develop mathematical foundations and computational workflows in which trainable representations learn missing mechanisms, interpretable tools explain them, and reliability analysis quantifies when the learned mechanism can be trusted.