A Physics-Informed Deep Learning Method With Adaptively Weighted Loss for Modeling Soil Water Flows

Richards' equation, widely used to model soil water flows, presents numerical challenges due to the high nonlinearity of its constitutive relationships. The deep learning method with a physics-informed neural network (PINN) provides a fresh perspective for solving this equation without prior knowledge of soil water constitutive relationship. However, existing PINN-based methods for Richards' equation are still significantly limited by the feasible soil types, and further developments are urgently needed to make the neural network models practically applicable to various types of soils. In this paper, we introduce a deep learning method, “PINN-AWL,” which simultaneously build the PINN model for prediction of soil water flows and establish the constitutive relationships between soil matric potential, soil water content and unsaturated hydraulic conductivity. An adaptively weighted loss is specially designed for the training process of this model. Specifically, the loss function is adjusted with self-adaptive weights at the training points in each iteration of training. This allows the proposed PINN-AWL to automatically focus more on regions where the solution is difficult to fit. The prediction accuracy and generalization ability of the PINN-AWL are thoroughly tested on various soils ranging from silty to sandy types. We also conduct studies to investigate the optimal structure and hyper-parameters used in the proposed method. The numerical results demonstrate that the proposed PINN-AWL significantly outperforms both the standard PINN and the monotonic PINN, especially on soils exhibiting strong nonlinearity in constitutive relationships, as indicated by larger “n” values in the van Genuchten-Mualem model.