Towards new soil water flow equations using physics-constrained machine learning

The Richardson-Richards equation (RRE) is a widely used partial differential equation (PDE) for modeling moisture dynamics in unsaturated soil. However, field soil moisture observations do not always satisfy RRE. In this paper, we introduce a new physically constrained machine learning (PCML) approach to derive governing soil water flow PDE directly from moisture observations. This paper is viewed as a feasibility study and reports results of our first attempt in developing the PCML approach. Here, we rely on noisy synthetic soil moisture data obtained from the linear RRE subject to real flux boundary conditions. The linear RRE was used as a reference PDE to check the PCML-derived PDEs, where the PCML performance was shown to be highly dependent upon the sample size in time and space. Results presented here confirm the feasibility of deriving soil water flow governing PDEs directly from soil moisture observations using PCML.