Authors: Yalan Song, Chaopeng Shen – Pennsylvania State University
Title: When ancient numerical demons meet physics-informed machine learning: Impacts of numerical accuracy on adjoint-based differentiable modeling
Abstract: The accurate prediction of hydrologic variables is crucial for effective water resource management. Recent developments in automatic differentiation (AD)-based in differentiable modeling (a genre of physics-informed machine learning) have enabled training intermingled neural networks to provide regionalized parameterization and process substitutes for process-based hydrological models. However, current models solve ordinary differential equations of process-based models discretely and explicitly, incurring large numerical errors, adding uncertainty to the learned parameters. Implicit numerical solutions with AD can result in gradient vanishing and excessive memory consumption, while existing adjoint-state methods can lead to inaccurate gradients. Here we propose adjoint methods, defined at several function levels, to increase computational efficiency and improve numerical accuracy. We demonstrate that the adjoint method can lead to better hydrologic simulation results than previous state-of-the-art differentiable model. In addition, the parameters obtained from a numerically more accurate model can be in different regions in the parameter space than the discrete version. Defining the adjoint at the equation level appears to introduce noticeable noise in this case. This work represents an early application of adjoint method in large-scale hydrologic applications.