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

Data-driven scientific discovery methods have been developed and applied to discover governing equations from data, involving the attempt to discover the unsaturated flow equation in soils from data. However, an important but unresolved problem is how to reconstruct the unsaturated flow equation from highly noisy and scarce discrete data. In this study, we present a new deep-learning framework: DeepGS (deep-learning-based group sparsity framework), that leverages the synergy of group sparsity and physics-informed deep learning (PIDL) to reconstruct the latent governing equation for unsaturated flow. In particular, we design a strategy that decomposes the identification of the unsaturated flow equation into two tasks: the determination of the partial differential equation structure and the reconstruction of the nonlinear coefficients. The tasks can be seamlessly handled by group sparse regression and the PIDL approach. Through the training, it realizes the simultaneous reconstruction of soil moisture dynamics and unsaturated flow governing equation. A series of comprehensive numerical experiments are conducted to determine the optimal architecture and test its performance. The results show the efficacy and robustness of DeepGS, which significantly outperform previous methods. We also conclude that accurately reconstructing soil moisture dynamics and spatiotemporal derivatives from noisy and scarce data play a critical role in governing equation discovery. This study further demonstrates the potential of discovering the governing equation for unsaturated flow from data in more complex scenarios, where rich and accurate soil moisture observations are generally intractable to access. Plain Language Summary Establishing an equation to describe soil water flow is important for scientists and engineers to understand its physical characteristics and apply it to scientific and engineering practice. Deriving the equation from physical principles step by step is difficult due to the complexity of soil water flow and even may be inaccurate. Recently, a class of physics-informed data-driven methods has been proposed, which enables learning the physical equations directly from data, and it has been applied to soil water flow equation establishment. However, it requires rich and accurate soil water observations, which are generally difficult to access. Here, we propose a new deep-learning approach to reduce the high dependence of previous methods on high-quality data. Specifically, we designed a special deep-learning architecture and its training method to realize this objective. We designed and conducted comprehensive numerical experiments to test the methods. This study provides insights into how to accurately discover the soil water flow equation from data. Broadly, it is a step forward in revealing yet unclear physical laws of soil water flow from data.