重拾基本原理：一种可微框架用于概念性水文模型
Reclaiming First Principles: A Differentiable Framework for Conceptual Hydrologic Models
AI总结：
本文提出了一种基于精确参数灵敏度的可微框架，用于高效且稳定地校准概念性水文模型。
摘要：
概念性水文模型仍然是降雨-径流建模的基石，但其校准往往速度缓慢且数值上容易出错。大多数基于梯度的参数估计方法依赖于有限差分近似或自动微分框架（如JAX、PyTorch和TensorFlow），这些方法计算上耗费资源，并引入截断误差、求解器不稳定性以及显著的开销。这些限制在概念性流域模型的ODE系统中尤为突出。在这里，我们介绍了一种完全解析且计算高效的可微水文建模框架，该框架基于精确的参数灵敏度。通过在支配的ODE系统中加入灵敏度方程，我们共同演化模型状态和相对于所有参数的雅可比矩阵。这个雅可比矩阵为任何可微损失函数提供完全解析的梯度向量。这些包括经典的客观函数，如绝对和平方残差之和，广泛使用的水文性能指标，如纳什-苏特里奇效率和金斯-古塔效率，稳健的损失函数，用于降低极端事件的权重，以及基于水文图的函数，如流量持续时间和退水曲线。解析灵敏度消除了数值微分中固有的步长依赖性和噪声，同时避免了共轭方法的不稳定性以及现代机器学习自动微分工具链的开销。最终的梯度是确定性的、物理上可解释的，并且易于嵌入基于梯度的优化器中。总体而言，这项工作使概念性水文模型的基于梯度的校准变得快速、稳定和透明，从而在不依赖外部、不透明或CPU密集型自动微分库的情况下释放可微建模的全部潜力。
英文摘要：
Conceptual hydrologic models remain the cornerstone of rainfall-runoff modeling, yet their calibration is often slow and numerically fragile. Most gradient-based parameter estimation methods rely on finite-difference approximations or automatic differentiation frameworks (e.g., JAX, PyTorch and TensorFlow), which are computationally demanding and introduce truncation errors, solver instabilities, and substantial overhead. These limitations are particularly acute for the ODE systems of conceptual watershed models. Here we introduce a fully analytic and computationally efficient framework for differentiable hydrologic modeling based on exact parameter sensitivities. By augmenting the governing ODE system with sensitivity equations, we jointly evolve the model states and the Jacobian matrix with respect to all parameters. This Jacobian then provides fully analytic gradient vectors for any differentiable loss function. These include classical objective functions such as the sum of absolute and squared residuals, widely used hydrologic performance metrics such as the Nash-Sutcliffe and Kling-Gupta efficiencies, robust loss functions that down-weight extreme events, and hydrograph-based functionals such as flow-duration and recession curves. The analytic sensitivities eliminate the step-size dependence and noise inherent to numerical differentiation, while avoiding the instability of adjoint methods and the overhead of modern machine-learning autodiff toolchains. The resulting gradients are deterministic, physically interpretable, and straightforward to embed in gradient-based optimizers. Overall, this work enables rapid, stable, and transparent gradient-based calibration of conceptual hydrologic models, unlocking the full potential of differentiable modeling without reliance on external, opaque, or CPU-intensive automatic-differentiation libraries.
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