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

Abstract

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.

Data‐Driven Discovery of Soil Moisture Flow Governing Equation: A Sparse Regression Framework

“…If feasible, data‐driven approaches would help to reduce problems, such as model structure errors (Q. Zhang et al., 2019) and complex parameter calibrations (Farthing & Ogden, 2017; Zha, Yang, et al., 2019), encountered by traditional simplified (Richards, 1931; Richardson, 1922) and highly parameterized (Brooks & Corey, 1964; Kosugi, 1994; Van Genuchten, 1980) equations. Related studies have been conducted on linear saturated flow (Chang & Zhang, 2019a, 2019b) and linear unsaturated flow (Ghorbani et al., 2021) and have achieved promising results. In this work, we further advance these works and propose a data‐driven framework for discovering the nonlinear time‐dependent soil moisture flow governing equation.…”

Reconstructing the Unsaturated Flow Equation From Sparse and Noisy Data: Leveraging the Synergy of Group Sparsity and Physics‐Informed Deep Learning

“…Here, we conduct an experiment to compare the performance of different methods to calculate the derivatives, including the spline used by Ghorbani et al. (2021), the DNN used by Song et al. (2022), and DeepGS proposed in this study.…”

“…Here, we conduct an experiment to compare the performance of different methods to calculate the derivatives, including the spline used by Ghorbani et al (2021), the DNN used by Song et al (2022), and DeepGS proposed in this study. Here, the derivatives calculated from clean data using the finite difference method are used as the correct reference values.…”

“…Nevertheless, an important but unresolved problem is that considerable observational data is required in current algorithms and is quite sensitive to measurement errors (Ghorbani et al., 2021; Song et al., 2022). However, measuring soil moisture data is still expensive and generally collected sparsely with measurement errors in field conditions (e.g., B. Li et al., 2022).…”

“…Luckily, there have been preliminary attempts to develop adapted routes. Ghorbani et al (2021) assumed a linearized partial differential equation (PDE) describes soil moisture dynamics and recovers it from data. However, the nonlinearity of governing equation is ignored, which is the essential feature of unsaturated flow.…”