Physics-Constrained Neural Reservoirs: A Powerful Neural Replacement of Conceptual Hydrological Laws for Learning Spatially Distributed Flow Dynamics

We introduce physics-constrained neural reservoirs (PyNaR), a hybrid spatially distributed rainfall–runoff framework in which flow dynamics are governed by neural operators, replacing traditional empirical laws. These operators enforce physical constraints via shape functions while remaining flexible, allowing the model to learn water fluxes driving reservoir dynamics directly from data. Prior knowledge is incorporated by pre-training neural reservoirs on synthetic fluxes from a hydrological model, which provides a physically consistent initialization before learning from observed discharge. Tested on challenging catchments across diverse hydroclimatic conditions, PyNaR demonstrates robustness and accuracy, outperforming classical conceptual, simpler hybrid, and purely data-driven LSTM models. Analysis of internal fluxes and response surfaces reveals distinct spatiotemporal patterns and realistic nonlinear behaviors learned by neural reservoirs, while achieving a better overall water balance. PyNaR offers a modular framework that can integrate new data and could be adapted to other geophysical models, enabling neural replacement within dynamic systems.