Physics‐Informed Neural Networks With Monotonicity Constraints for Richardson‐Richards Equation: Estimation of Constitutive Relationships and Soil Water Flux Density From Volumetric Water Content Measurements

Abstract Water retention curves (WRCs) and hydraulic conductivity functions (HCFs) are critical soil‐specific characteristics necessary for modeling the movement of water in soils using the Richardson‐Richards equation (RRE). Well‐established laboratory measurement methods of WRCs and HCFs are not usually suitable for simulating field‐scale soil moisture dynamics because of the scale mismatch. Hence, the inverse solution of the RRE must be used to estimate WRCs and HCFs from field measured data. Here, we propose a physics‐informed neural network (PINN) framework for the inverse solution of the RRE and the estimation of WRCs and HCFs from only volumetric water content (VWC) measurements. The proposed framework does not need initial and boundary conditions, which are rarely available in real applications. The PINNs consist of three linked feedforward neural networks, two of which were constrained to be monotonic functions to reflect the monotonicity of WRCs and HCFs. Alternatively, we also tested PINNs without monotonicity constraints. We trained the PINNs using synthetic VWC data with artificial noise, derived by a numerical solution of the RRE for three soil textures. The monotonicity constraints regularized the inverse problem. The PINNs were able to reconstruct the true VWC dynamics. We demonstrated that the PINNs could recover the underlying WRCs and HCFs in nonparametric form. However, the reconstructed WRCs and HCFs at wet and dry ends were unsatisfactory because of the strong nonlinearity. We additionally showed that the trained PINNs could estimate soil water flux density with a broader range of estimation than the currently available methods.