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.

中文翻译：

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使用物理约束机器学习建立新的土壤水流方程

Richardson-Richards 方程 (RRE) 是一种广泛使用的偏微分方程 (PDE)，用于模拟非饱和土壤中的水分动力学。然而，田间土壤水分观测并不总是满足 RRE。在本文中，我们引入了一种新的物理约束机器学习 (PCML) 方法，以直接从水分观测中推导出控制土壤水流 PDE。本文被视为一项可行性研究，并报告了我们首次尝试开发 PCML 方法的结果。在这里，我们依赖于从受实际通量边界条件影响的线性 RRE 获得的嘈杂的合成土壤水分数据。线性 RRE 被用作参考 PDE 来检查 PCML 派生的 PDE，其中 PCML 性能被证明高度依赖于时间和空间的样本大小。