Inverse Modelling of Shallow Land Subsidence using a Hybrid Differentiable Physics-Machine Learning Approach

Shallow subsidence is a key problem in many coastal plains, such as those of the Mississippi (U.S.A), the Mekong (Vietnam), Po (Italy), and Rhine deltas (Netherlands). Managing it is as important as managing anticipated sea-level rise. Accurate prediction of shallow land subsidence into the coming century, requires robust parameterization of a series of complexly interacting physical and biochemical processes (leading to consolidation, oxidation, shrinkage), that operate across heterogeneous shallow subsurface conditions. Traditional physics-based models depend on parameters that are difficult to constrain spatially, while purely data-driven approaches might give physically inconsistent results and often lack physical interpretability. We present a hybrid modeling framework that balances this tradeoff by combining a fully differentiable version of the process-based subsidence model Atlantis with neural network components for learning spatial parameter heterogeneity, enabling gradient-based parameter optimization directly from observational data.

Our approach converts established isotach consolidation model and peat-oxidation calculation methods into a differentiable computational graph, allowing automatic differentiation to propagate observational constraints through the physics model. This enables joint inversion of spatially distributed InSAR-derived observations and vertical extensometer profiles to constrain process parameters at voxel scale. Crucially, we incorporate observation uncertainty (the full variance-covariance structure) of InSAR-derived measurements through a statistically rigorous loss function (Mahalanobis distance), properly accounting for spatial and temporal correlations that traditional calibration approaches neglect. First results confirm that the framework can recover peat oxidation parameters from synthetic subsidence observations; integration of InSAR-derived data with full uncertainty characterization is underway.

The differentiable architecture offers several advantages: 1) principled uncertainty quantification by accounting for the error structure of input observations, 2) efficient optimization through gradient descent rather than computationally expensive sampling methods, and 3) flexible integration of heterogeneous data sources within a unified modeling framework. We demonstrate the approach spatially, using various observations for a long-managed mainly agricultural peat meadow polder (Krimpenerwaard, The Netherlands; current average elevation 1.5 m below MSL and sinking). Our methodology bridges geodetic remote sensing with process-based geotechnical modeling, contributing to improved projections of coastal relative sea-level rise by constraining subsurface processes at operationally relevant scales. The approach is computationally efficient and can be scaled to larger areas and longer timeframes, depending on the availability of novel InSAR-derived observations and subsurface data.