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

Abstract

AbstractData‐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.