A hybrid physics–AI approach using universal differential equations with state-dependent neural networks for learnable, regionalizable, spatially distributed hydrological modeling

Abstract. Conceptual hydrological models, traditionally relying on simplified representations of physical processes governed by conservation laws 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. In parallel, 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 framework that embeds neural networks (NNs) seamlessly into a spatialized, regionalizable, and fully differentiable process-based model via universal differential equations (UDEs). The model integrates a state-dependent NN to refine internal water fluxes and an implicit resolution of the UDE system, followed by kinematic wave routing on a flow direction grid. Spatially distributed parameters are inferred through regionalization mappings including convolutional NNs, and adjoint-based gradients enable end-to-end training of the hybrid system. We implement this framework into the latest release of the smash platform, significantly extending its capabilities to comprehensively evaluate hybrid models at kilometric spatial and hourly temporal resolutions. The results show that hybrid approaches demonstrate consistently strong and stable performance in calibration and various validation scenarios. Additionally, the UDE structure exhibits a hybridization effect that modifies state dynamics and runoff flow, achieving more accurate streamflow simulations for flood modeling.