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A Hybrid Neural Network-Finite Element Method for the Viscous-Plastic Sea-Ice Model

Nils Margenberg, Carolin Mehlmann

TL;DR

The paper tackles the computational bottleneck of solving the viscous-plastic sea-ice VP model by coupling a coarse finite element discretization with patch-local neural network corrections learned from high-resolution data. The Hybrid NN-FEM framework enriches the coarse solution in a refined auxiliary space, applying localized, batched neural corrections after a Newton-Krylov solve to accelerate convergence and better capture linear kinematic features without a full fine-grid solve. Extensive experiments show that, on an 8 km grid, the method achieves accuracy and deformation statistics close to a 4 km reference, with Newton iterations and runtime substantially reduced and neural overhead remaining negligible. The work demonstrates robust, scalable, and locality-driven learning-based enhancements for multiscale climate simulations, with ablations highlighting the importance of both state- and residual-information in the network inputs. Overall, NN-FEM offers a practical path to faster, high-fidelity sea-ice modeling on coarse grids, enabling more efficient climate simulations while preserving key deformation patterns.

Abstract

We present an efficient hybrid Neural Network-Finite Element Method (NN-FEM) for solving the viscous-plastic (VP) sea-ice model. The VP model is widely used in climate simulations to represent large-scale sea-ice dynamics. However, the strong nonlinearity introduced by the material law makes VP solvers computationally expensive, with the cost per degree of freedom increasing rapidly under mesh refinement. High spatial resolution is particularly required to capture narrow deformation bands known as linear kinematic features in viscous-plastic models. To improve computational efficiency in simulating such fine-scale deformation features, we propose to enrich coarse-mesh finite element approximations with fine-scale corrections predicted by neural networks trained with high-resolution simulations. The neural network operates locally on small patches of grid elements, which is efficient due to its relatively small size and parallel applicability across grid patches. An advantage of this local approach is that it generalizes well to different right-hand sides and computational domains, since the network operates on small subregions rather than learning details tied to a specific choice of boundary conditions, forcing, or geometry. The numerical examples quantify the runtime and evaluate the error for this hybrid approach with respect to the simulation of sea-ice deformations. Applying the learned network correction enables coarser-grid simulations to achieve qualitatively similar accuracy at approximately 11 times lower computational cost relative to the high-resolution reference simulations. Moreover, the learned correction accelerates the Newton solver by up to 10% compared to runs without the correction at the same mesh resolution.

A Hybrid Neural Network-Finite Element Method for the Viscous-Plastic Sea-Ice Model

TL;DR

The paper tackles the computational bottleneck of solving the viscous-plastic sea-ice VP model by coupling a coarse finite element discretization with patch-local neural network corrections learned from high-resolution data. The Hybrid NN-FEM framework enriches the coarse solution in a refined auxiliary space, applying localized, batched neural corrections after a Newton-Krylov solve to accelerate convergence and better capture linear kinematic features without a full fine-grid solve. Extensive experiments show that, on an 8 km grid, the method achieves accuracy and deformation statistics close to a 4 km reference, with Newton iterations and runtime substantially reduced and neural overhead remaining negligible. The work demonstrates robust, scalable, and locality-driven learning-based enhancements for multiscale climate simulations, with ablations highlighting the importance of both state- and residual-information in the network inputs. Overall, NN-FEM offers a practical path to faster, high-fidelity sea-ice modeling on coarse grids, enabling more efficient climate simulations while preserving key deformation patterns.

Abstract

We present an efficient hybrid Neural Network-Finite Element Method (NN-FEM) for solving the viscous-plastic (VP) sea-ice model. The VP model is widely used in climate simulations to represent large-scale sea-ice dynamics. However, the strong nonlinearity introduced by the material law makes VP solvers computationally expensive, with the cost per degree of freedom increasing rapidly under mesh refinement. High spatial resolution is particularly required to capture narrow deformation bands known as linear kinematic features in viscous-plastic models. To improve computational efficiency in simulating such fine-scale deformation features, we propose to enrich coarse-mesh finite element approximations with fine-scale corrections predicted by neural networks trained with high-resolution simulations. The neural network operates locally on small patches of grid elements, which is efficient due to its relatively small size and parallel applicability across grid patches. An advantage of this local approach is that it generalizes well to different right-hand sides and computational domains, since the network operates on small subregions rather than learning details tied to a specific choice of boundary conditions, forcing, or geometry. The numerical examples quantify the runtime and evaluate the error for this hybrid approach with respect to the simulation of sea-ice deformations. Applying the learned network correction enables coarser-grid simulations to achieve qualitatively similar accuracy at approximately 11 times lower computational cost relative to the high-resolution reference simulations. Moreover, the learned correction accelerates the Newton solver by up to 10% compared to runs without the correction at the same mesh resolution.

Paper Structure

This paper contains 37 sections, 46 equations, 9 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Variational Formulation.
  4. Discretization in Space and Time.
  5. Discretization of the Sea-Ice Momentum Equation
  6. Hybrid Neural Network-Finite Element Method
  7. Level Hierarchy and Notation
  8. The Hybrid Algorithm
  9. Neural Network Inputs
  10. Inputs.
  11. Output and Assembly.
  12. Patch Localization of Inputs, Batched Prediction, and Global Assembly
  13. Gather.
  14. Network Input.
  15. Patch-wise Prediction.
  16. ...and 22 more sections

Figures (9)

  • Figure 1: Compact overview of one hybrid NN–FEM time step $t^{n}\to t^{n+1}$: (1) use the corrected state $\mathbf{v}_\text{fine}^{n}$ from the previous step to assemble the fine-space right-hand side $\mathbf f_\text{fine}^{n+1}$ and (2) restrict it to the working space, $\mathbf f_\text{coarse}^{n+1} = \mathbf R\mathbf f_\text{fine}^{n+1}$, (3) solve the sea-ice momentum equations on $\mathcal{V}_\text{coarse}$ to obtain $\mathbf{v}_\text{coarse}^{n+1}$, (4) prolongate $\mathbf{v}_\text{coarse}^{n+1}$ to the auxiliary fine space, $\mathbf{v}_\text{fine}^{n+1} = \mathbf P\mathbf{v}_\text{coarse}^{n+1} \in \mathcal{V}_\text{fine}$, (5) apply a patch-wise neural-network correction based on the residual and state information to compute a fine-scale increment $\delta\mathbf{v}_\text{fine}^{n+1} \in \mathcal{V}_\text{fine}$, and store the corrected fine-grid state $\mathbf{v}_\text{fine}^{n+1} = \mathbf{v}_\text{fine}^{n+1} + \delta\mathbf{v}_\text{fine}^{n+1}$ for use in assembling $\mathbf f_\text{fine}^{n+2}$ at the beginning of the next time step.
  • Figure 2: Patch geometry and notation. Thin gray lines: coarse mesh $\mathcal{T}_\text{coarse}$. Dashed lines: auxiliary refinement on $\mathcal{T}_\text{fine}$ restricted to a patch. A single-cell patch $M_{K}$ on $\mathcal{T}_\text{coarse}$ collects the $4^{S}$ child patches on $\mathcal{T}_\text{fine}$ corresponding to a single coarse cell $K \in \mathcal{T}_\text{coarse}$ (cf. panel (a) and panel (b). A multi-cell patch is the union over a set of cells $K$, $\mathcal{B}\subset\mathcal{T}_\text{coarse}$ (cf. panel (c)).
  • Figure 3: Batched input matrix composed of per-patch (one patch = one row) velocity entries $\mathcal{Q}(\mathbf{v}_\text{fine})$, residual entries $\mathcal{Q}(\mathbf{r}_\text{fine})$, and geometric descriptors $\boldsymbol\omega$; the network $\mathcal{N}$ maps each row (patch) to a vector of patch-local corrections $\delta\mathbf{v}^{M}_{K}$.
  • Figure 4: Test and training setups. For training the model we consider the sea-ice benchmark problem defined in Mehlmann2021, where a cyclone is moving in northeast direction over an ice-covered domain. Panel (a) shows sea-ice shear deformation after 4 days of simulation. Panel (b) and (c) indicates the wind field of the cyclone $\mathbf{v}_{\mathrm w}$ at $t=1d$ and $t=3d$, respectively. For testing the network the wind field are either anti-cyclone or cyclone moving in north-west (NW), south-west (SW) or south-east (SE) direction. The different directions are indicated in gray in panel b and c.
  • Figure 5: Shear deformation simulated with the benchmark setup, where the direction of the cyclone is changed to north-west. The top row shows the simulations performed without NN: (a) 8km coarse resolution simulation, (b) 4km reference simulation, the target for NN-FEM with $S=1$, (c) 2km reference simulation, the target for NN-FEM with $S=2$. The middle and bottom rows show the simulated shear deformation derived from a NN-FEM simulation on $\mathcal{V}_\text{coarse}$ with a mesh size of 8km. The corresponding NN uses varying patch size $N_M$ and operates on $\mathcal{V}_\text{fine}$ with mesh sizes of 4km, i.e. $S=1$ (middle row) and 2km, i.e. $S=2$ (bottom row).
  • ...and 4 more figures

Theorems & Definitions (7)

  • Remark 3.1
  • Remark 4.1: On the role of Geometric Multigrid in the hybrid NN-FEM approach
  • Remark 4.2: Impact of the correction on Newton's method
  • Remark 4.3: Neural network design for stability
  • Remark 4.4: Parallelism, batching, and conservation
  • Remark 5.1: NN-FEM approach on large patches
  • Remark A.1: Impact of the correction on inexact Newton admissibility