Physics-informed neural networks for Richardson-Richards equation:
Estimation of constitutive relationships and soil water flux density
from volumetric water content measurements
Abstract
Water retention curve (WRC) and hydraulic conductivity function (HCF)
are essential information to model the movement of water in the soil
using the Richardson-Richards equation (RRE). Although laboratory
measurement methods of WRC and HCF have been well established, the
lab-based WRC and HCF can not be used to model soil moisture dynamics in
the field because of the scale mismatch. Therefore, it is necessary to
derive the inverse solution of the RRE and estimate WRC and HCF from
field measurement data. We are proposing a physics-informed neural
networks (PINNs) framework to obtain the inverse solution of the RRE and
estimate WRC and HCF from only volumetric water content measurements.
The PINNs was constructed using three feedforward neural networks, two
of which were constrained to be monotonic functions to reflect the
monotonicity of WRC and HCF. The PINNs was trained using noisy synthetic
volumetric water content data derived from the simulation of soil
moisture dynamics for three soils with distinct textures. The PINNs
could reconstruct the true soil moisture dynamics from the noisy data.
As for WRC, the PINN could not precisely determine the WRCs. However, it
was shown that the PINNs could estimate the HCFs from only the noisy
volumetric water content data without specifying initial and boundary
conditions and assuming any information about the HCF (e.g., saturated
hydraulic conductivity). Additionally, we showed that the PINNs
framework could be used to estimate soil water flux density with a
broader range of estimation than the currently available methods.