One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections of sea-level rise, considering the uncertainty in ice-sheet dynamics. However, accurate quantification of the uncertainty requires running the ice-sheet models for a large number of parameter samples, which is often infeasible due to the cost of ice-sheet computational models. To address this issue, we propose a hybrid approach to approximate existing finite-element ice-sheet models at a fraction of their cost. In this approach, the finite-element model for the momentum equations, which is the most expensive part of an ice-sheet model, is replaced by a Deep Operator Network, while the classic finite-element discretization for the evolution of the ice thickness is retained. We show that the resulting hybrid model is accurate and an order of magnitude faster than the traditional finite-element model. A distinctive feature of the proposed model is that it can handle high-dimensional parameter spaces, e.g., the basal friction field at the bed of the glacier, and, therefore, it can be used for efficiently evaluating the model for different samples of the parameters.

We demonstrate our approach targeting the evolution of the Humboldt Glacier in Greenland. We initialize our finite-element model using simulation-constrained optimization to solve for the basal friction field that minimizes the misfit with observed velocities in 2007. We then generate samples of the basal friction from a Gaussian process centered at the optimal basal friction field. For each sample we run the model forward in time and use the model-generated velocity and thickness data to train our deep-learning surrogate. We show that our hybrid model can provide statistics of the glacier mass loss that are in agreement with those computed using the reference finite-element model. For increased accuracy, the hybrid model can be effectively used in a multi-fidelity strategy where a relatively small number of evaluations of the finite-element model (considered the high-fidelity model) are performed together with evaluations of the hybrid model to improve the projections of the glacier mass loss.