Hybrid Physics-AI and Neural ODE Approaches for Spatially Distributed Hydrological Modeling

Abstract. Empirical models are among the earliest hydrological models and have evolved from the unit hydrograph to deep learning models. Despite their success, purely data-driven methods often lack interpretability and are highly sensitive to data quality, limiting their generalizability in data-scarce regions or under changing environmental conditions. Conceptual models, traditionally relying on simplified representations of physical processes governed by conservation laws of mass, momentum, and energy, remain widely used in operational hydrology due to their explainability and practical applicability. However, these process-based models inherently face structural uncertainties and a lack of scale-relevant theories—challenges that emerging artificial intelligence (AI) techniques may help address. Moreover, high-resolution models are crucial for predicting extreme events characterized by strong variability and short duration, making spatially distributed hybrid modeling critical in the current context. We introduce a hybrid physics-AI approach that integrates neural ordinary differential equations (ODEs), solved by an implicit numerical scheme, into a spatialized, regionalizable, and differentiable process-based model. The hydrological module is built on a continuous state-space system and an integrated process-parameterization neural network. This hybrid system solves the ODEs governing reservoir dynamics, while embedding a neural network to refine internal water fluxes, all without relying on an analytical solution, instead computing the model states simultaneously. This work also presents an upgraded version of the smash platform following its initial release, featuring a more comprehensive evaluation of hybrid models at relatively fine resolutions of kilometric spatial and hourly temporal scales. The results show that hybrid approaches demonstrate consistently strong and stable performance in calibration and various validation scenarios. Additionally, the neural ODE structure exhibits a hybridization effect that modifies state dynamics and runoff flow, achieving more reliable streamflow simulations for flood modeling.