PINN-HYDRO: A Temporally Adaptive Physics-Informed Neural Networks Framework for Subsurface Hydrologic Predictions Under Varying Boundary and Initial Conditions

Physics-Informed Neural Networks (PINNs) have shown promise in hydrologic predictions by approximating physics-driven Partial Differential Equations (PDEs) using observed data, particularly in subsurface hydrology where observations are limited. However, traditional PINNs lack generalizability, requiring retraining for different initial and boundary conditions. This study presents PINN-HYDRO, an enhanced temporal PINN framework for predicting soil moisture profiles based on Richard's PDE of water mass conservation in unsaturated soil.

Our approach incorporates meteorological forcing variables (temperature, solar radiation, wind speed, and rainfall) alongside traditional Richard's PDE domain variables (time and depth) to generalize the model for soil moisture profile prediction under various boundary forcing scenarios. To address lagged correlations between rainfall and soil moisture, we introduced cumulative rainfall over various time windows as additional inputs. An attention mechanism was implemented to modulate the influence of these inputs based on their relevance to different boundary conditions.

The model was developed and validated using a year-long dataset with 5-minute resolution, collected from four sensors located in Austin, Texas: one meteorological and three soil moisture sensors at increasing depths (33cm to 77cm). To assess transferability, we tested the trained model on Texas Soil Observation Network's (TxSON) soil sensors data.

During the testing phase, PINN-HYDRO significantly outperformed traditional PINN in predicting soil moisture profiles, particularly during and after extreme rainfall events. The correlation between predicted and observed soil moisture at all three depths improved from 0.63 to 0.97, while the Mean Square Error (MSE) decreased from 3x10^-3 to 1x10^-4.

Results demonstrated PINN-HYDRO's enhanced ability to adapt to diverse boundary conditions and capture complex temporal dynamics in subsurface hydrology. This approach can potentially improve the accuracy and applicability of traditional PINNs used in subsurface hydrology.