Hybrid training for robust emulator-based ice-thickness inversion

Ice-flow inversions aim to infer unobserved controls on glacier and ice-sheet dynamics from limited, noisy surface data but are notoriously ill-posed: multiple parameter fields can reproduce the same observations, solutions are sensitive to priors/regularization, and model nonlinearity amplifies both data and structural errors. Here we target a particularly challenging variant — emulator-based inversion using a machine-learning surrogate for ice flow — where the forward operator is fast and differentiable but only an approximation of the governing physics. We focus on inverting for ice thickness, which remains poorly constrained for most glaciers yet strongly conditions driving stress, basal traction, and therefore hindcast skill and projection uncertainty.

We present emulator-based inversions with the Instructed Glacier Model (IGM), benchmarking against synthetic tests with known truth and contrasting performance with a full-physics ice-flow solver. IGM provides a PINN-based emulator trained by minimizing a energy representing the Blatter–Pattyn equations. This powerful approach has proven very successful in the forward problem but leads to an emulator that may need regular retraining to ensure an accurate solution. We show that this training approach can introduce surrogate error modes that distort gradients and create spurious minima, degrading convergence and reliability of gradient-based optimization used for the inverse problem. To address this, we introduce a hybrid training strategy that augments the physics loss with a data-misfit term against a large training set, with the aim of improving out-of-distribution generalization across glacier geometries. The resulting emulator yields more reliable recovery of unknown fields such as ice thickness and supports the fast, scalable inversions needed for ensemble modelling and robust uncertainty quantification.