Cheap and accurate higher-order ice flow emulator for mountain glaciers

Abstract. Numerical projections of mountain-glacier evolution increasingly rely on large ensembles that sample uncertain climate forcing, geometry, and ice-flow parameters, yet explicitly resolving higher-order ice dynamics within such ensembles remains computationally expensive. Here we take an alternative approach by training a neural network once, offline, to reproduce the velocity field of a higher-order ice-flow model, and then use the trained network in place of the ice-flow solver during transient simulations. In doing so, we retain the higher-order model's physical basis while avoiding the repeated cost of its velocity solve. The network predicts depth-dependent horizontal ice-flow velocities for land-terminating mountain glaciers directly from gridded glacier geometry and physical parameters, and its weights are then held fixed, so that no retraining is required when it is applied to a new glacier. Leveraging the computational efficiency of GPU operations together with the instructed glacier model (IGM), we generate a large synthetic training set of transient glacier states across diverse mountain topography and climate histories, and at fixed intervals we compute reference velocities from the Blatter-Pattyn ice-flow formulation. The resulting dataset contains over 300,000 higher-order velocity solutions, orders of magnitude more than the catalogues used to train earlier ice-flow emulators. We train on the misfit between the network outputs and target velocities, together with a physics term based on the same discretised ice-flow energy. Our preferred architecture, trained with this hybrid objective, accurately reproduces velocity on examples unseen during training, with a median surface-speed error of 0.71 m yr^-1 and a median flux-divergence error of 0.10 m yr^-1. In transient simulations of two real glacier systems that lie outside the training set, the emulator tracks reference simulation ice volume through a 500-year advance-retreat cycle with maximum errors below 6%. Crucially, we find that errors do not accumulate over time. Instead, velocity biases induce geometric changes that tend to counteract them, stabilising the coupled emulator–mass-conservation system. Taken together, our results show that the expensive part of higher-order ice flow can be paid for once during training and reused indefinitely: velocity is predicted in a single forward pass rather than solved for iteratively at every time step. Higher-order ice dynamics are therefore no longer the limiting cost in mountain-glacier modelling, and become practical in large regional and global projection ensembles.