A General Framework for Integrating Neural Networks into Numerical Resolution Methods for Spatially Distributed Hydrological Models

Machine learning (ML) methods have been utilized in hydrology for decades. Recently, hybrid approaches that combine data-driven techniques with process-based models have gained attention, highlighting the complementary strengths of ML and physical models. However, the explicability and adaptability of such hybrid models remain open questions. This work introduces a general framework for incorporating neural networks (NNs) and ML techniques into a regionalizable, spatially distributed hydrological model. As a case study, a simple NN is employed to correct internal fluxes within a conceptual GR hydrological model that allows analytical integration. The corresponding hybrid ordinary differential equation set is integrated with an implicit numerical scheme solved by the Newton-Raphson method. Implementation in Fortran-based code supports differentiability, enabling the computation of the cost gradient through a combination of an adjoint model and analytical NN gradients. Results over a large catchment sample show promising improvements in model accuracy and provide insights into hydrological behaviors through interpretable NN outputs. These findings demonstrate the framework's potential to advance hybrid hydrological modeling by enhancing explicability and adaptability. Additionally, the proposed framework offers flexibility for integration into other modeling chains and applications across diverse geophysical models.