Identification of a soil water flow equation using a two-step data-driven method

Abstract

Governing equations of soil water flow are traditionally derived from the conservation of mass and the Buckingham-Darcy law. However, due to uncertainties in many aspects, such as physical processes and constitutive relationships, a thorough understanding of soil water flow in a theory-driven way often poses many challenges. With the fast improvement of data acquisition ability, data-driven approaches have gained considerable scientific interest in discovering the governing equations of physical problems. However, most existing studies focused on linear or weakly nonlinear problems with constant coefficients in their partial differential equations. This study attempted to propose a modified two-step data-driven method to identify the strongly nonlinear equation of unsaturated flow. The Richardson-Richards equation and coefficients were jointly identified by hybridizing sparse regression and iterative smoother. The automatic differentiation scheme was introduced to reduce the risk of performance collapse from data noise common in sparse regression methods. We demonstrated the ability, and the challenge as well, of identifying the soil water flow governing equations in a data-driven way with the aid of a series of synthetic cases. Our results showed that the modified method was capable of accurately identifying the governing equation of unsaturated flow, whose performance was almost independent of prior knowledge of the equation form and constitutive relationships. If the underlying physical processes are poorly understood, then a more relaxed equation form was recommended. The reduction in the spatiotemporal resolution of sampling may cause higher-order terms to be difficult to derive accurately or some terms that were not originally present to appear in the identified equation. Although not completely eliminated, the detrimental effect of data noise on the identification of the governing equations can be greatly mitigated by introducing the automatic differentiation scheme.