The Effect of Different Optimization Strategies to Physics-Constrained Deep Learning for Soil Moisture Estimation

Soil moisture is a key hydrological parameter that has significant importance to human society and the environment. Accurate modeling and monitoring of soil moisture in crop fields, especially in the root zone (top 100 cm of soil), is essential for improving agricultural production and crop yield with the help of precision irrigation and farming tools. Realizing the full sensor data potential depends greatly on advanced analytical and predictive domain-aware models. In this work, we propose a physics-constrained deep learning (P-DL) framework to integrate physics-based principles on water transport and water sensing signals for effective reconstruction of the soil moisture dynamics. We adopt three different optimizers, namely Adam, RMSprop, and GD, to minimize the loss function of P-DL during the training process. In the illustrative case study, we demonstrate the empirical convergence of Adam optimizers outperforms the other optimization methods in both mini-batch and full-batch training.

Physics-constrained Active Learning for Soil Moisture Estimation and Optimal Sensor Placement

Soil moisture is a crucial hydrological state variable that has significant importance to the global environment and agriculture. Precise monitoring of soil moisture in crop fields is critical to reducing agricultural drought and improving crop yield. In-situ soil moisture sensors, which are buried at pre-determined depths and distributed across the field, are promising solutions for monitoring soil moisture. However, high-density sensor deployment is neither economically feasible nor practical. Thus, to achieve a higher spatial resolution of soil moisture dynamics using a limited number of sensors, we integrate a physics-based agro-hydrological model based on Richards' equation in a physics-constrained deep learning framework to accurately predict soil moisture dynamics in the soil's root zone. This approach ensures that soil moisture estimates align well with sensor observations while obeying physical laws at the same time. Furthermore, to strategically identify the locations for sensor placement, we introduce a novel active learning framework that combines space-filling design and physics residual-based sampling to maximize data acquisition potential with limited sensors. Our numerical results demonstrate that integrating Physics-constrained Deep Learning (P-DL) with an active learning strategy within a unified framework--named the Physics-constrained Active Learning (P-DAL) framework--significantly improves the predictive accuracy and effectiveness of field-scale soil moisture monitoring using in-situ sensors.

A Data-facilitated Numerical Method for Richards Equation to Model Water Flow Dynamics in Soil

Root-zone soil moisture monitoring is essential for precision agriculture, smart irrigation, and drought prevention. Modeling the spatiotemporal water flow dynamics in soil is typically achieved by solving a hydrological model, such as the Richards equation which is a highly nonlinear partial differential equation (PDE). In this paper, we present a novel data-facilitated numerical method for solving the mixed-form Richards equation. This numerical method, which we call the D-GRW (Data-facilitated global Random Walk) method, synergistically integrates adaptive linearization scheme, neural networks, and global random walk in a finite volume discretization framework to produce accurate numerical solutions of the Richards equation with guaranteed convergence under reasonable assumptions. Through three illustrative examples, we demonstrate and discuss the superior accuracy and mass conservation performance of our D-GRW method and compare it with benchmark numerical methods and commercial solver.