Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheets

TL;DR

The paper addresses the computational bottleneck of uncertainty quantification for large-scale ice-sheet PDEs by developing a physics-inspired graph neural network surrogate that operates on unstructured meshes. It combines a Hamiltonian-bracket GNN architecture with graph attention to preserve physical structure while mitigating oversmoothing, and introduces transfer learning and domain decomposition to achieve fast, accurate training on massive graphs. Key findings show substantial training-time reductions and improved predictive accuracy when employing TL and DD, with notable gains near the glacier terminus and promising avenues for UQ foundation-models. The work demonstrates a scalable, data-efficient path to reliable surrogate modeling for ice-sheet dynamics and other PDE-governed systems, with broad applicability to uncertainty quantification tasks.

Abstract

Accurate yet efficient surrogate models are essential for large-scale simulations of partial differential equations (PDEs), particularly for uncertainty quantification (UQ) tasks that demand hundreds or thousands of evaluations. We develop a physics-inspired graph neural network (GNN) surrogate that operates directly on unstructured meshes and leverages the flexibility of graph attention. To improve both training efficiency and generalization properties of the model, we introduce a domain decomposition (DD) strategy that partitions the mesh into subdomains, trains local GNN surrogates in parallel, and aggregates their predictions. We then employ transfer learning to fine-tune models across subdomains, accelerating training and improving accuracy in data-limited settings. Applied to ice sheet simulations, our approach accurately predicts full-field velocities on high-resolution meshes, substantially reduces training time relative to training a single global surrogate model, and provides a ripe foundation for UQ objectives. Our results demonstrate that graph-based DD, combined with transfer learning, provides a scalable and reliable pathway for training GNN surrogates on massive PDE-governed systems, with broad potential for application beyond ice sheet dynamics.

Figures (8)

• Figure 1: Dual Delaunay triangulation used as the graph representation of the Humboldt Glacier. In the MALI ice sheet model, ice thickness and temperature are discretized using a finite-volume method on a Voronoi mesh, while the velocity equations are solved on its dual Delaunay triangulation using low-order Lagrangian finite elements. We use this dual mesh as the node–edge structure for our GNN surrogate. Colors indicate cell area, with red indicating large cell area and blue indicating small cell area. Note that cell area decreases toward the glacier terminus where velocities and uncertainty are greatest, corresponding to finer mesh resolution in the finite element simulations.
• Figure 2: Sample realizations of the input features for our GNN surrogate, plotted on the Humboldt Glacier. Left: bed topography [m]; Center: basal friction [yr/m]$^{q}$, where the value of $q$ accounts for the nonlinearity of the sliding law; Right: ice thickness [m].
• Figure 3: Overview of training pipeline, with different strategies considered. Phase 0 is an optional pre-training step that utilizes work already completed during model development, or abundant data pertaining to another related graph (e.g. a subgraph of the ice sheet of interest, or a completely different ice sheet).
• Figure 4: Subdomains produced by spectral clustering algorithm, for $k=3$.
• Figure 5: Comparison of three training strategies on the same test snapshot. Top row: ground-truth velocity magnitude (full domain and zoomed terminus view). Rows 2-4 show: (i) cold start, (ii) warm start (pre-trained on a subgraph and fine-tuned globally), and (iii) warm start plus DD, where subdomain models are pre-trained, fine-tuned, and stitched at inference. Columns display the magnitude of pointwise velocity error, full-field velocity prediction, and zoomed-in velocity prediction. The error color scale is intentionally not fixed across rows to highlight improvements: the overall magnitude of error decreases substantially from cold start to warm start to warm start plus DD, demonstrating progressively stronger accuracy, particularly in the terminus region.