KAN-GCN: Combining Kolmogorov-Arnold Network with Graph Convolution Network for an Accurate Ice Sheet Emulator

AI summary

Overview

Research area: Machine learning for climate science, specifically graph neural network emulators of numerical ice-sheet models (a paper submitted to the New Perspectives in Graph Machine Learning workshop, arXiv:2510.24926v1 [cs.LG]).

Technical level: Intermediate. Readers need some familiarity with graph neural networks and basic neural network training concepts, but the paper's core idea is explained without requiring deep background in glaciology or PDE solvers.

Scope: The paper proposes KAN-GCN, a hybrid architecture that places a Kolmogorov–Arnold Network encoder in front of graph convolutional layers, and benchmarks it as an emulator for Pine Island Glacier ice-sheet simulations across three mesh resolutions and 2–5 network depths.

What This Paper Is About

Numerical ice-sheet models solve coupled nonlinear partial differential equations describing how glaciers flow and thin, which makes them accurate but computationally expensive. Prior work showed that graph convolutional networks (GCNs) can emulate these models on the irregular triangular meshes that the physics solvers use, but those emulators still produced large errors, especially for ice velocity. This paper asks whether inserting a Kolmogorov–Arnold Network as a per-feature nonlinear encoder before the GCN stack — plus a reformulated prediction target and a re-weighted loss — can improve accuracy while keeping inference fast.

Key Contributions

1. A KAN-first architecture. The authors place a Kolmogorov–Arnold Network as a node-wise feature encoder ahead of the graph convolution stack. The KAN applies learnable univariate functions (B-splines in the general formulation, Gaussian radial basis functions in the FastKAN variant used here) plus a linear mix, adding feature-wise nonlinearity without increasing the number of message-passing hops.

2. A reformulated learning target. Instead of predicting absolute ice thickness and velocity at time t directly from the initial state, the emulator is trained to predict the one-step residual updates Δv_t = v_t − v_{t−1} and ΔH_t = H_t − H_{t−1}, which are then added back to the previous state. The authors argue this shortens the temporal gap, improves conditioning, and aligns with finite-difference update rules in physical solvers.

3. Feature curation and a feature-wise weighted loss. The input representation is pruned from the prior work's feature set to six channels (melting rate, surface mass balance, time, V_x, V_y, thickness), dropping velocity magnitude, surface/base elevation, and the ice-floating field. Training uses a scale-weighted MSE loss with separate coefficients λ_v and λ_h for velocity and thickness.

4. A systematic baseline comparison. KAN-GCN is compared against a pure GCN emulator and an MLP–GCN baseline at matched depths from 2 to 5 layers, evaluated on accuracy, parameter count, and total GPU inference time for producing the full 20-year transient simulation with 36 melting rates.

Main Findings

• Residual reformulation drives the largest accuracy gains. Relative to the 5GCN baseline of Koo et al. (thickness RMSE 12.9629 on the 2 km mesh, 13.9422 on 5 km, 14.9061 on 10 km), switching to the residual target dropped thickness RMSE to 9.0505 / 10.8579 / 11.7124. Adding feature curation brought it to 0.9212 / 0.9467 / 0.9616, and loss re-weighting reduced it further.

• KAN-GCN is more accurate than pure GCN at depths of 3–5 layers, but slightly worse at 2 layers. For example, thickness RMSE for KAN+4GCN (0.3850 / 0.3923 / 0.4245 on the 2 km / 5 km / 10 km meshes) beats the same-depth 5GCN (0.3901 / 0.4046 / 0.4376). KAN+3GCN (0.4043 / 0.4061 / 0.4321) beats 4GCN (0.4119 / 0.4188 / 0.4535). But at 2 total layers, KAN+GCN (0.5996 / 0.5919 / 0.6314) is worse than 2GCN (0.4790 / 0.4773 / 0.5158). The authors explain this via message-passing depth: replacing a GCN layer with a KAN front end leaves only one hop of aggregation in a 2-layer stack.

• KAN-GCN's advantage is much larger for velocity than for thickness. Velocity RMSE for KAN+3GCN is 2.0407 / 2.1897 / 2.0159 versus 4GCN's 2.1913 / 2.3661 / 2.1485, and KAN+2GCN is 2.3612 / 2.4314 / 2.2767 versus 3GCN's 2.6371 / 2.7371 / 2.5394. The authors attribute this to velocity being more nonlinear and anisotropic than thickness, so KAN's adaptive univariate bases capture feature-specific responses more effectively.

• One corner case favors the MLP front end. On velocity with the 5-layer architecture and the 10 km mesh, MLP+4GCN achieves RMSE 1.9696 versus KAN+4GCN's 1.9789. The authors suggest that with fewer nodes and edges plus deep aggregation, a lighter MLP regularizes better and avoids the mild overfitting a higher-capacity KAN can induce.

• KAN-GCN is also more accurate than the MLP–GCN baseline at every depth tested. For thickness, KAN+GCN (0.5996 / 0.5919 / 0.6314) beats MLP+GCN (0.6980 / 0.6898 / 0.7269); for velocity, KAN+4GCN (1.9212 / 2.0364 / 1.9789) beats MLP+4GCN (2.0452 / 2.2094 / 1.9696) on two of the three meshes.

• Parameter overhead is small and constant. The KAN front end adds about 6.2k parameters at every depth: 23959 versus 17795 parameters at 2 layers, 40471 versus 34307 at 3 layers, 56983 versus 50819 at 4 layers, and 73495 versus 67331 at 5 layers.

• Inference is faster on coarser meshes despite the extra parameters. For example, KAN+3GCN takes 3.6947 seconds versus 4GCN's 4.4377 seconds on the 10 km mesh, and 3.447 versus 4.0523 seconds on the 5 km mesh. On the finest mesh (2 km), KAN+3GCN takes 5.4000 seconds versus 4GCN's 5.0967 seconds — a modest cost. The authors attribute the speedup to swapping an edge-heavy message-passing layer for a dense node-wise transform, which reduces launch and synchronization overhead that dominates on coarser (smaller) graphs.

Methodology in Plain English

The team ran numerical ice-sheet simulations with the Ice-sheet and Sea-level System Model (ISSM) for the Pine Island Glacier in Antarctica, using the Shelfy-Stream Approximation. They produced 20-year simulations of ice thickness and velocity under 36 different annual basal-melt scenarios ranging from 0 to 70 m/a in 2 m/a increments, each run at three mesh resolutions (2 km, 5 km, and 10 km) — 108 simulations in total, yielding 25,812 graphs (239 residual values × 3 mesh sizes × 36 melting rates). The graph structure comes directly from the ISSM triangular mesh: mesh nodes become graph nodes and mesh edges become graph edges. Each node carries six input features — melting rate, time t−1, surface mass balance at t−1, V_x and V_y at t−1, and thickness at t−1 — and the model predicts V_x, V_y, and thickness at the next timestep via the residual formulation. All inputs and outputs are normalized to [−1, 1] using nominal feature ranges, following the prior work.

Data were split by melting rate: 0, 20, 40, and 60 m/a for validation, and 10, 30, 50, and 70 m/a for testing, giving 20,076 training graphs, 2,868 validation graphs, and 2,868 test graphs. Splitting by melt rate means the test set probes out-of-distribution melting conditions.

Three architectures were compared at depths from 2 to 5 layers: pure GCN, MLP–GCN (an MLP encoder instead of KAN), and KAN–GCN. All were trained on the same hardware (AMD EPYC Milan processor, 120 GB system RAM, one NVIDIA A100 GPU with 40 GB VRAM) using the Adam optimizer. Baselines without the re-weighted loss used a learning rate of 0.01 with an exponential scheduler (gamma 0.99); the advanced emulators used a learning rate of 0.0005 with a cosine annealing scheduler. All models were trained for 500 epochs, implemented with the Deep Graph Library (DGL) and PyTorch, and used Leaky ReLU with a negative slope of 0.01 as the activation in the GCN stack.

Why This Matters

Ice sheets in Greenland and Antarctica are major contributors to sea-level rise, and the numerical models used to project their future behavior are computationally expensive enough to limit how many scenarios can be explored. A fast, accurate emulator makes large scenario sweeps practical. This paper's contribution is showing that where you put the nonlinearity in the architecture — feature-wise before aggregation, rather than only inside message passing — matters for both accuracy and throughput.

Real-world applications:

• Sea-level rise projection. Faster emulators allow many more melt-rate and climate scenarios to be evaluated, which supports ensembles of projections rather than single runs.
• Model calibration and uncertainty quantification. Because emulators are cheap to evaluate, they can be used inside calibration loops that would be infeasible with the full physics solver.
• Glacier-specific hazard and process studies. The same approach could be applied to other fast-changing glaciers where velocity and thickness evolution need to be tracked under varying melt conditions.
• Climate adaptation and infrastructure planning. Long-horizon coastal planning depends on the range of plausible sea-level outcomes, which broader scenario sweeps can better characterize.

Industry relevance: the work sits at the intersection of GPU-accelerated scientific computing and climate analytics. Techniques for swapping edge-heavy graph operations for node-wise transforms are relevant beyond glaciology to any domain using graph neural networks on large irregular meshes, and the emulator approach is relevant to climate risk modeling, insurance, and infrastructure engineering.

Future Directions

• Test generalization beyond Pine Island Glacier. All experiments are on one glacier; whether the KAN-first design transfers to other Antarctic or Greenland glaciers is not reported.
• Improve KAN-GCN at shallow depth. The 2-layer case underperforms pure GCN, and the authors connect this to reduced message-passing depth. Architectures that preserve both feature-wise nonlinearity and sufficient aggregation remain an open question.
• Address the finest-mesh overhead. The per-node spline/basis evaluations of the KAN layer are not fully amortized on the 2 km mesh, where the model shows a modest cost versus pure GCN; the authors do not report a fix.
• Investigate the 5-layer, 10 km velocity corner case. The paper attributes the MLP's advantage there to regularization effects and possible mild KAN overfitting, but does not test that hypothesis directly.
• Explore the interpretability claim further. The paper states that KAN's univariate functions reveal variable effects for key physical drivers such as melt rate, surface mass balance, and temperature, but does not report specific learned response curves or their physical interpretation.

Target Audience

This paper is most useful to machine learning researchers working on neural surrogates and graph neural networks for physical simulation, and to computational glaciologists and climate scientists looking for faster emulators of ice-sheet models. It is also relevant to practitioners interested in architecture design for mesh-based scientific data, particularly those weighing accuracy against inference throughput across different mesh resolutions. Readers without background in graph neural networks or ice-sheet modeling will find the architecture description accessible but the experimental framing demanding.

Authors’ abstract

We introduce KAN-GCN, a fast and accurate emulator for ice sheet modeling that places a Kolmogorov-Arnold Network (KAN) as a feature-wise calibrator before graph convolution networks (GCNs). The KAN front end applies learnable one-dimensional warps and a linear mixing step, improving feature conditioning and nonlinear encoding without increasing message-passing depth. We employ this architecture to improve the performance of emulators for numerical ice sheet models. Our emulator is trained and tested using 36 melting-rate simulations with 3 mesh-size settings for Pine Island Glacier, Antarctica. Across 2- to 5-layer architectures, KAN-GCN matches or exceeds the accuracy of pure GCN and MLP-GCN baselines. Despite a small parameter overhead, KAN-GCN improves inference throughput on coarser meshes by replacing one edge-wise message-passing layer with a node-wise transform; only the finest mesh shows a modest cost. Overall, KAN-first designs offer a favorable accuracy vs. efficiency trade-off for large transient scenario sweeps.