Physics-Coupled and Message-Transferred Inverse Modeling for Subsurface Flow with Very Sparse Supervision

Inverse modeling for subsurface flow represents a fundamental scientific challenge in hydrogeology and geotechnical engineering, which seeks to reconstruct critical hydrogeological parameters from sparse observational constraints. The marked spatial heterogeneity of subsurface formations, combined with the prohibitively high costs of data acquisition, renders parameter inversion, especially with very sparse supervision, inherently ill-posed and susceptible to non-uniqueness and instability. Numerical simulation-based iterative inversion methods are computationally expensive and time-consuming. Purely data-driven approaches require extensive labeled data, whereas the existing physics-informed methods lack an explicit architecture-level information transfer channel between parameter and response fields. Under sparse supervision, this prevents hydraulic head observations from effectively constraining hydraulic conductivity identification, resulting in weak parameter identifiability. In this work, we propose a physics-coupled and message-transferred inverse modeling method for transient subsurface flow problems with very sparse supervision. Specifically, the static parameter field estimated by the inversion network is explicitly incorporated into the dynamic response prediction network, and the static inversion and dynamic prediction networks are physics-coupled by the governing equations in parallel. This method enables accurate hydraulic conductivity inversion under extremely limited supervision. Experiments on multiple parameter fields, label scales, and noise levels demonstrate accurate and stable inversion performance under very sparse supervision, with ensemble-based uncertainty analysis, further confirming the reliability of the proposed method.