Exact and scalable gradient-based learning of conceptual hydrologic models using analytic forward sensitivities

Gradient-based methods are increasingly used in hydrologic model calibration, data assimilation, and hybrid physics–machine learning frameworks. However, most existing approaches rely on finite differences, automatic differentiation, or surrogate emulators, which are computationally expensive, memory-intensive, and sensitive to numerical noise, especially for long time series and nontraditional objective functions. We present a general framework for exact, scalable gradient computation in conceptual hydrologic models based on analytic forward sensitivity equations. By augmenting the governing ODEs with sensitivity states, a single model integration simultaneously yields hydrologic states, fluxes, and the full parameter Jacobian. These sensitivities are independent of the objective function, enabling exact gradients for any differentiable loss, including least squares, absolute residuals, NSE, KGE, flow-duration-curve metrics, and robust M-estimators, without re-running the model or invoking automatic differentiation. We implement this approach in a suite of widely used conceptual models (including HBV, HYMOD, Hmodel, GR4J, SAC-SMA, and Xinanjiang) within a unified computational framework with a high-performance C++ core and MATLAB/Python interfaces. We demonstrate its scalability using a large-sample experiment based on the CAMELS data set, comprising 671 catchments across the contiguous United States. Compared to automatic and numerical differentiation, our approach reduces calibration times from hours to minutes while improving numerical stability, convergence behavior, and interpretability. This work establishes analytic forward sensitivities as a transparent, physics-consistent, and computationally efficient foundation for large-sample hydrology and process-based model learning.