A Spatiotemporal Physics-Motivated State-Space Model of Lake Temperature Profiles

The physical structure of lakes, particularly vertical temperature stratification, governs the dissolved oxygen distribution and underpins freshwater ecosystem health. Accurately predicting temperature profiles is therefore essential for understanding key ecological processes. Traditional mechanistic models provide physically consistent simulations but can be computationally intensive and sensitive to input assumptions, whereas purely data-driven models may offer high predictive accuracy, yet lack physical interpretability. In this work, we present a mixture-of-experts framework that integrates a physics-motivated neural network with an unconstrained deep neural network model to predict lake water temperatures across depths. The physics-motivated component enforces physically consistent vertical profiles via a learnable logistic formulation, while the data-driven expert captures empirical patterns from observations. A gating mechanism dynamically combines the two experts, balancing statistical flexibility and physical fidelity. Applied across lakes with diverse morphologies, the framework accurately reproduces the timing, magnitude, and vertical structure of thermal stratification. The model is further used to project future lake temperatures under climate change scenarios, which point to a warming trend in near-surface temperatures, providing a tool to assess potential shifts in stratification and their ecological consequences for freshwater systems.