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 unsuitable for simulating field-scale soil moisture dynamics because of the scale mismatch. Hence, the inverse solution of the RRE is used to estimate WRCs and HCFs from field measured data. Here, we propose a physics-informed neural networks (PINNs) framework for the inverse solution of the RRE and the estimation of WRCs and HCFs from only volumetric water content (VWC) measurements. Unlike conventional inverse methods, the proposed framework does not need initial and boundary conditions. The PINNs consists 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 PINNs were able to reconstruct the true VWC dynamics. The monotonicity constraints prevented the PINNs from overfitting the training data. We demonstrated that the PINNs could recover the underlying WRCs and HCFs in non-parametric form, without a need for initial guess. However, the reconstructed WRCs at near-saturation–which was not fully represented in the training data–was unsatisfactory. We additionally showed that the trained PINNs could estimate soil water flux density with a broader range of estimation than the currently available methods.

Potential evapotranspiration (ETp), defined as the amount of water that would be lost by evaporation and transpiration from an area in the absence of water limitation, is an import hydrometeorological variable. Accurate estimation ETp is critical for a wide range of applications including predictions of, irrigation water requirement, groundwater recharge, stream discharge, drought and wildfires. Long-term change in ETp is considered an indicator of the impact of climate change on ecosystem functioning. A wide range of physically-based and empirical models have been developed to estimate ETp. These methods can be explained in terms of their complexity data requirements. The most complex and demanding models (e.g., Penmann-Montheith) require measurements of radiation, temperature, windspeed, and vapor pressure and have been shown to provide very close approximation of physically measured ET from unstressed systems. On the other extreme, the simplest models require only temperature data (e.g., Thornthwaite, 1948) and are the most commonly implemented. However, without site-specific calibration, methods that depend solely on temperature achieve only modest accuracy. Here we present a machine-learning (ML) approaches that utilize hourly and sub-hourly temperature records to produce predictions that are comparable with the more complex methods that require full meteorological datasets. Specifically, we show that ML algorithms can learn the patterns of temperature fluctuations that are related to attenuation of potential solar radiation. We anticipate the approach developed here will be valuable for estimation of historical ETp as well as for short-term forecasting using temperature forecasts.