Predicting the future contribution of the ice sheets to sea level rise over the next decades presents several challenges due to a poor understanding of critical boundary conditions, such as basal sliding. Traditional numerical models often rely on data assimilation methods to infer spatially variable friction coefficients by solving an inverse problem, given an empirical friction law. However, these approaches are not versatile, as they sometimes demand extensive code development efforts when integrating new physics into the model. Furthermore, this approach makes it difficult to handle sparse data effectively. To tackle these challenges, we propose a novel approach utilizing Physics-Informed Neural Networks (PINNs) to seamlessly integrate observational data and governing equations of ice flow into a unified loss function, facilitating the solution of both forward and inverse problems within the same framework. We illustrate the versatility of this approach by applying the framework to two-dimensional problems on the Helheim Glacier in southeast Greenland. By systematically concealing one variable (e.g. ice speed, ice thickness, etc.), we demonstrate the ability of PINNs to accurately reconstruct hidden information. Furthermore, we extend this application to address a challenging mixed inversion problem. We show how PINNs are capable of inferring the basal friction coefficient while simultaneously filling gaps in the sparsely observed ice thickness. This unified framework offers a promising avenue to enhance the predictive capabilities of ice sheet models, reducing uncertainties, and advancing our understanding of poorly constrained physical processes.

At least half of today’s mass loss of the Greenland ice sheet is due to the retreat of tidewater glaciers. For example, over the past decade Helheim Glacier in southeast Greenland has been one of the largest contributors to total ice discharge across the Greenland ice sheet. There is broad agreement that the acceleration and retreat of these marine terminating glaciers has been triggered by the intrusion of warmer currents in the fjords, however, other processes such as changes in basal conditions, ice rheology, surface mass balance or calving dynamics may have also played important roles in controlling the retreat of these glaciers. Without quantifying the individual contributions of these processes, it is difficult to determine which of these processes should be included in ice sheet models to correctly capture the present and future retreat and associated mass loss of the ice sheet. In this study, we simulate the dynamics of Helheim Glacier, from 2007 to 2020, using the Ice-sheet and Sea-level System Model (ISSM) to investigate the model response to changes in external forcing and boundary conditions. By switching off each of these external forcing components and comparing the numerical solution with observations, we identify that the seasonal to inter-annual variability of Helheim Glacier is relatively insensitive to the choice of friction law or the ice rheology, but that the position of the calving front has a direct and large impact on ice velocity.We then apply automatic differentiation to quantify the transient sensitivity of the ice flux near the terminus to changes in ocean-induced melting rates, basal frictions, ice rheology, calving dynamics and surface mass balance. These sensitivities highlight the regions where each parameter may contribute the most to changes in ice flux and which process should be properly captured by numerical models in order to accurately project the future response of Helheim Glacier. This study, as a result, can be used as a guide for model development of similar glaciers.