Grounded ice thickness plays a critical role in understanding the behavior and stability of ice sheets, particularly in polar regions such as Greenland, Antarctica, and the Canadian Arctic. Ice sheet dynamics are governed by complex interactions between ice flow, surface accumulation, and bedrock topography, making the accurate modeling of these processes essential for predicting long-term ice sheet behavior and their contributions to global sea level rise [14, 18]. In particular, the Shallow Ice Approximation (SIA) provides a framework for modeling grounded ice, where ice flow is driven by internal deformation and the base is often assumed to be frozen, constraining the ice thickness by bedrock topography [12, 15]. A key challenge in modeling grounded ice involves solving the partial differential equations (PDEs) that govern ice thickness evolution, while incorporating these constraints. This leads to a free boundary problem, where the ice thickness must remain non-negative and above the bedrock, giving rise to an obstacle problem [21, 3].

The Ice-sheet and Sea-level System Model (ISSM) provides solutions for Stokes equations relevant to ice sheet dynamics by employing finite element and fine mesh adaption. However, since its finite element method is compatible only with Central Processing Units (CPU), the ISSM has limits on further economizing computational time. Thus, by taking advantage of Graphics Processing Units (GPUs), we design a graph convolutional network (GCN) as a fast emulator for ISSM. The GCN is trained and tested using the 20-year transient ISSM simulations in the Pine Island Glacier (PIG). The GCN reproduces ice thickness and velocity with a correlation coefficient greater than 0.998, outperforming the traditional convolutional neural network (CNN). Additionally, GCN shows 34 times faster computational speed than the CPU-based ISSM modeling. The GPU-based GCN emulator allows us to predict how the PIG will change in the future under different melting rate scenarios with high fidelity and much faster computational time.

Although numerical models provide accurate solutions for ice sheet dynamics based on physics laws, they accompany intensified computational demands to solve partial differential equations. In recent years, convolutional neural networks (CNNs) have been widely used as statistical emulators for those numerical models. However, since CNNs operate on regular grids, they cannot represent the refined meshes and computational efficiency of finite-element numerical models. Therefore, instead of CNNs, this study adopts an equivariant graph convolutional network (EGCN) as an emulator for the ice sheet dynamics modeling. EGCN reproduces ice thickness and velocity changes in the Helheim Glacier, Greenland, and Pine Island Glacier, Antarctica, with 260 times and 44 times faster computation time, respectively. Compared to the traditional CNN and graph convolutional network, EGCN shows outstanding accuracy in thickness prediction near fast ice streams by preserving the equivariance to the translation and rotation of graphs.

The mass loss of the polar ice sheets contributes considerably to ongoing sea-level rise and changing ocean circulation, leading to coastal flooding and risking the homes and livelihoods of tens of millions of people globally. To address the complex problem of ice behavior, physical models and data-driven models have been proposed in the literature. Although traditional physical models can guarantee physically meaningful results, they have limitations in producing high-resolution results. On the other hand, data-driven approaches require large amounts of high-quality and labeled data, which is rarely available in the polar regions. Hence, as a promising framework that leverages the advantages of physical models and data-driven methods, physics-informed machine learning (PIML) has been widely studied in recent years. In this paper, we review the existing algorithms of PIML, provide our own taxonomy based on the methods of combining physics and data-driven approaches, and analyze the advantages of PIML in the aspects of accuracy and efficiency. Further, our survey discusses some current challenges and highlights future opportunities, including PIML on sea ice studies, PIML with different combination methods and backbone networks, and neural operator methods.

Although the finite element approach of the Ice-sheet and Sea-level System Model (ISSM) solves ice dynamics problems governed by Stokes equations quickly and accurately, such numerical modeling requires intensive computation on central processing units (CPU). In this study, we develop graph neural networks (GNN) as fast surrogate models to preserve the finite element structure of ISSM. Using the 20-year transient simulations in the Pine Island Glacier (PIG), we train and test three GNNs: graph convolutional network (GCN), graph attention network (GAT), and equivariant graph convolutional network (EGCN). These GNNs reproduce ice thickness and velocity with better accuracy than the classic convolutional neural network (CNN) and multi-layer perception (MLP). In particular, GNNs successfully capture the ice mass loss and acceleration induced by higher basal melting rates in the PIG. When our GNN emulators are implemented on graphic processing units (GPUs), they show up to 50 times faster computational time than the CPU-based ISSM simulation.

One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections of sea level rise. A large part of the uncertainty of sea level projections is due to uncertainty in ice sheet dynamics. At the moment, accurate quantification of the uncertainty is hindered by the cost of ice sheet computational models. In this work, we develop a hybrid approach to approximate existing ice sheet computational models at a fraction of their cost. Our approach consists of replacing the finite element model for the momentum equations for the ice velocity, the most expensive part of an ice sheet model, with a Deep Operator Network, while retaining a classic finite element discretization for the evolution of the ice thickness. We show that the resulting hybrid model is very accurate and it is an order of magnitude faster than the traditional finite element model. Further, a distinctive feature of the proposed model compared to other neural network approaches, is that it can handle high-dimensional parameter spaces (parameter fields) such as the basal friction at the bed of the glacier, and can therefore be used for generating samples for uncertainty quantification. We study the impact of hyper-parameters, number of unknowns and correlation length of the parameter distribution on the training and accuracy of the Deep Operator Network on a synthetic ice sheet model. We then target the evolution of the Humboldt glacier in Greenland and show that our hybrid model can provide accurate statistics of the glacier mass loss and can be effectively used to accelerate the quantification of uncertainty.

Ice thickness estimation is an important aspect of ice sheet studies. In this work, we use convolutional neural networks with multiple output nodes to regress and learn the thickness of internal ice layers in Snow Radar images collected in northwest Greenland. We experiment with some state-of-the-art networks and find that with the residual connections of ResNet50, we could achieve a mean absolute error of 1.251 pixels over the test set. Such regression-based networks can further be improved by embedding domain knowledge and radar information in the neural network in order to reduce the requirement of manual annotations.

Today, large resources are used to provide vessels in the polar seas with warnings about the spread of sea ice. Artificial intelligence may make these warnings cheaper, faster, and available for everyone. For vessels that journey into the polar seas, keeping control of the spread of sea ice is critical, which means that large resources are spent to collect data and determine future developments to provide reliable sea ice warnings. "As of now, large resources are needed to create these ice warnings, and most of them are made by The Norwegian Meteorological Institute and similar centers," says Sindre Markus Fritzner, a doctoral research fellow at UiT The Arctic University of Norway. He is employed at the Department of Physics and Technology and has recently submitted a doctoral thesis in which he looked at the option of using artificial intelligence to make ice warnings faster, better, and more accessible than they are today.

For vessels that journey into the polar seas, keeping control of the spread of sea ice is critical, which means that large resources are spent to collect data and determine future developments to provide reliable sea ice warnings. "As of now, large resources are needed to create these ice warnings, and most of them are made by The Norwegian Meteorological Institute and similar centres", Sindre Markus Fritzner tells us. He is a Doctoral Research Fellow at UiT The Arctic University of Norway. Fritzner is employed at the Department of Physics and Technology and has recently submitted a doctoral thesis where he has looked at the option of using artificial intelligence to make ice warnings faster, better, and more accessible than they are today. The ice warnings used today are traditionally based on dynamic computer models that are fed with satellite observations of the ice cover, and whatever updated data can be gathered about ice thickness and snow depth.