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

Abstract

The dynamics of soil moisture in the vadose zone are mathematically modeled by a nonlinear partial differential equation (PDE), which is referred to as the Richardson-Richards equation (RRE;Richards, 1931;Richardson, 1922). Conventionally, theoretical methods for deriving the RRE are rooted in physics laws, including the conservation of mass and the Buckingham-Darcy law (Buckingham, 1907) with conditional simplification assumptions (Narasimhan, 2007;Pinder & Celia, 2006). A thorough understanding of its physical processes is required to determine the soil moisture flow governing equation. However, due to the complexity and uncertainties in the vadose zone, such as the preferential flow through soil macropores (Beven & Germann, 2013;Nimmo, 2021), soil heterogeneity (Elkateb et al., 2003, and changing boundary conditions (Van Dam & Feddes, 2000), a physical principle-based rigorous derivation is complicated and even impossible. In addition, the applications of the RRE rely on accurate measurements of constitutive relationship model parameters, including the water retention curves and hydraulic conductivity functions (Madi et al., 2018). Nevertheless, the determination of soil hydraulic properties is time-consuming, laborious, and costly (Bitterlich et al., 2004). In addition, the experimental scale mismatch problem can worsen the results (Hopmans et al., 2002).With the continuous development of statistical machine learning algorithms, data-driven methods have achieved great success (Remesan & Mathew, 2015). Recently, a seminal breakthrough involved the identification of physical equations through data-driven approaches. Data-driven approaches aim to directly identify physical equations in ordinary differential equations or PDEs from the observed data. Related research traces back to Bongard and Lipson (2007) and Schmidt and Lipson (2009). They successfully used symbolic regression (Koza & Koza, 1992) to discover the underlying dynamic equations from captured motion-tracking data. Subsequently, more data-driven methods, including sparse regression (Brunton et al., 2016;Tibshirani, 1996) and inverse methods, such as Gaussian process regression (Raissi et al., 2017), data assimilation (Chang & Zhang, 2019a), and physics-informed neural networks (Raissi et al., 2019), have also been successfully applied to physical equation discovery.Considering these novel data-driven approaches have shown promising potential in discovering physics equations from data, we raise the following scientific question. Leveraging the powerful learning capability of data-driven methods, can the actual soil moisture flow governing equation be recovered from data, and can the automatic Abstract Conventionally, soil moisture dynamics are mathematically modeled by the Richardson-Richards equation, whose derivation is based on the conservation of mass and the Buckingham-Darcy law. However, it is complicated and even impossible to finish such rigorous derivations based on physical principles due to the complexity and uncertainties in the vadose zon...

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

“…The DeepGS approach is proposed to address the fact that previous methods (Song et al., 2022; we call it DNN‐GSR here) would fail in the case of highly noisy sparse data. The data availability and the measurement noise will affect the accuracy of the discovered unsaturated flow equation.…”

“…(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.…”

“…Therefore, we follow the method in Song et al. (2022) to realize the sparsity of coefficients $$\boldsymbol{\xi }(\theta )$$. The general idea is that considering the volumetric water content dependence in coefficients, when the variation of $$\theta $$ is small, the coefficients should remain essentially unchanged.…”

“…For instance, the maximum number of iterations 𝐴𝐴 𝐴𝐴gsr is set as 10 and threshold sequences number 𝐴𝐴 𝐴𝐴tol is set as 500, to allow the algorithm to heuristically locate an optimal threshold. The selection of these parameters is related to the Pareto front analysis and remains to be further studied (e.g., S. Rudy et al, 2019;Song et al, 2022).…”

“…This new perspective is more appropriate and practical for complex physical systems with elusive latent mechanisms (Z. Chen et al, 2021), including unsaturated flow modeling (Song et al, 2022). These advances are primarily attributed to the development of sparse regression in which parsimony governing terms with nonzero coefficients are identified from candidates (Kutz & Brunton, 2022).…”