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HI-MeshGraphNets: Efficient and Accurate Mesh-based Physics Learning with Hierarchical Multi-scale Graph Neural Networks

SiHun Lee, Dong-Hyuk Park, Taesoo Bang, Seung-Hoon Kang

arXiv:2608.13827v1cs.LG

TL;DR

Flat graph message passing limits long-range communication because information moves only one hop per layer. HI-MGN introduces a hierarchical extension of MeshGraphNets, improving accuracy across three benchmarks while reducing VRAM usage by up to 51.3% and training time by 63.0%.

  • Problem

    Flat graph message passing propagates information only one hop per layer, limiting long-range communication on meshes.

  • Method

    HI-MGN extends MeshGraphNets with hierarchical multiscale processing to address limited long-range information propagation.

  • Results

    Up to 51.3% lower VRAM usage and 63.0% shorter training time were achieved relative to computationally demanding baselines, with improved accuracy across three benchmarks.

  • Takeaways & Limitations

    HI-MGN provides a practical hierarchical approach for improving mesh-based physics surrogate modeling across the evaluated benchmarks.

  • Takeaways & Limitations

    The paper reports decreased performance in representing local geometry.

Abstract

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Machine-learned physical surrogate models have become promising alternatives to mesh-based numerical solvers. Among them, graph neural networks (GNNs) are well suited for representing simulation meshes and learning nodal state evolution through message passing. However, conventional flat message passing becomes inefficient on large, high-fidelity meshes because information propagates only one hop per layer, requiring deep processors for long-range interactions and increasing computational cost, memory usage, and the risk of over-smoothing. To address this limitation, we propose Hierarchical Interpolating MeshGraphNets (HI-MGN), a multiscale extension of MeshGraphNets for efficient long-range communication on unstructured meshes. HI-MGN replaces the flat processor with a hierarchical multiscale processor that coarsens graphs using farthest-point sampling and Voronoi partitioning while preserving the original mesh topology. Message passing on coarse graphs enables information to travel over larger geometric distances with fewer layers, and a learned graph interpolation network reconstructs fine-resolution features. Across three structural and fluid benchmarks, HI-MGN achieves improved accuracy compared with MeshGraphNets and the Bi-Stride Multi-Scale GNN while reducing training time and peak memory usage. The results show that topology-aware hierarchical message passing and learned coarse-to-fine interpolation provide an effective and practical framework for scalable mesh-based physics surrogate modeling.

1. Introduction

GNN-based mesh surrogates face memory growth and limited long-range interaction because standard message passing expands receptive fields by only one hop per layer. HI-MGN addresses these bottlenecks with a self-contained hierarchical graph construction and interpolation framework that improves accuracy while reducing training time and peak memory usage.

  • Motivation: GNN-based physics surrogates face excessive memory requirements and limited long-range interaction as mesh size increases.These bottlenecks remain particularly important on large graphs.
  • Motivation: Standard message passing propagates information one hop per layer, so distant interactions require deeper processors that cause over-smoothing and information attenuation.A node’s receptive field grows by a single hop per layer, while required steps scale with graph geodesic distance.
  • Prior multiscale methods: Existing multiscale methods mitigate long-range limitations but require prescribed coarse meshes or may need excessive hierarchical levels on very large meshes.MultiScale MGN requires pre-generated coarse meshes, whereas BSMS-GNN relies on repeated bi-stride coarsening.
  • HI-MGN: HI-MGN introduces a self-contained hierarchical graph construction and interpolation framework for mesh-based GNN surrogates.The framework is designed for efficient and accurate full-field prediction with enhanced long-range interaction.
  • HI-MGN: HI-MGN alleviates the long-range communication bottleneck with only a few message passing steps on coarse graphs.Its hierarchy supports strongly reduced coarse representations with user-specified node counts while preserving mesh-native modeling goals.
  • Results: HI-MGN achieves improved accuracy while substantially reducing training time and peak memory usage.These are reported outcomes of the proposed framework.

2. Methodology

HI-MGN extends MeshGraphNets with a hierarchical multiscale processor that enables long-range message passing over coarsened graphs. FPS–Voronoi coarsening preserves mesh topology, while learned geometry-aware interpolation reconstructs fine-resolution latent features.

  • Hierarchical multiscale processor: HI-MGN replaces the conventional flat processor with a hierarchical multiscale processor organized across H coarse graphs.This design makes long-range graph interactions tractable by passing messages at multiple resolutions.
  • Hierarchical multiscale processor: Coarse-level message passing shortens communication paths, allowing information to propagate farther with the same number of message-passing steps.The processor progressively pools latent states to coarser graphs, refines them, then upsamples the coarsest representation and merges skip states.
  • FPS–Voronoi coarsening: FPS–Voronoi coarsening selects seeds using farthest-point sampling and assigns nodes through multi-source breadth-first search over original mesh adjacency.Coarse nodes are FPS seeds, their features are mean-pooled from Voronoi clusters, and recursive application yields multiple hierarchical levels.
  • FPS–Voronoi coarsening: Boundary-induced coarse edges preserve mesh topology and avoid artificial connections between nearby but unconnected components.Two coarse nodes connect when a fine-mesh edge crosses their corresponding Voronoi clusters.
  • Learned interpolation: A learned geometry-aware interpolation GNN lifts coarse latents to fine resolution using neighboring coarse sources, skip connections, and reference-coordinate offsets.Aggregated coarse-to-fine messages are combined with the stored fine-scale skip connection to reconstruct fine latent states.

11: return z

The training setup standardizes channels using training-split statistics, optimizes a channel-weighted Huber loss, and applies robustness-oriented perturbations and symmetry-preserving augmentations. Training also uses several optimization and precision techniques, including AdamW, gradient clipping, scheduling, mixed precision, and EMA.

  • Inputs and outputs are standardized to zero mean and unit variance using statistics computed from the training split.
  • The network minimizes a channel-weighted Huber loss with normalized output-channel weights summing to one.The Huber loss grows linearly beyond its transition threshold, making it less sensitive to large residuals than mean squared error.
  • During training, perturbations are applied to temporal physical-state inputs, followed by recomputation of geometry-dependent edge features and consistent target correction.This procedure is intended to improve autoregressive rollout robustness and can address localized large errors near stress concentrations or contact regions.
  • Data augmentation includes random rotations and reflections when the governing physics and boundary conditions remain invariant under those transformations.
  • Optimization uses AdamW, gradient clipping, learning-rate warmup with cosine annealing, 16-bit mixed-precision training, and an exponential moving average of model weights.

3. Numerical examples

Across three benchmarks, HI-MGN consistently outperforms MeshGraphNets and BSMS-GNN in prediction accuracy while reducing training time and peak memory usage. These advantages persist in geometric extrapolation and challenging three-dimensional contact settings, whereas steady aerodynamics shows comparable accuracy with computational benefits.

  • Overall comparison: Across three benchmarks, HI-MGN consistently outperforms MeshGraphNets and BSMS-GNN while improving accuracy, training time, and peak memory usage.The comparison keeps MeshGraphNets and HI-MGN message-passing blocks identical, while BSMS-GNN uses seven to nine multiscale levels.
  • 2D thermoelastic example: HI-MGN produces smoother, more accurate rectangular-inclusion extrapolations than BSMS-GNN and MeshGraphNets, which poorly estimate the central region.MeshGraphNets captures boundary responses accurately but performs poorly in the central region.
  • 2D thermoelastic example: 38.6% lower peak VRAM usage and 59.9% lower training time are achieved by HI-MGN relative to the original MeshGraphNets baseline.BSMS-GNN uses 9.93GB VRAM and requires 6.48 hours of training.
  • 3D nonlinear contact example: HI-MGN outperforms existing methods in three-dimensional contact prediction and maintains better agreement than MeshGraphNets despite limited long-range propagation.The comparison includes MeshGraphNets with both 28 and 15 message-passing blocks, with contact onset examined at t = 12.
  • Steady aerodynamic example: HI-MGN achieves the highest R2 values for Cp, Cfx, and Cfy, while all models show comparable steady-aerodynamic accuracy and MeshGraphNets marginally leads for Cfz.Computational differences are more pronounced: HI-MGN uses the least peak VRAM, while BSMS-GNN trains faster.

4. Conclusions

HI-MGN extends MeshGraphNets with hierarchical, topology-aware message passing and learned geometry-aware interpolation to improve long-range communication on unstructured meshes. Across structural and aerodynamic benchmarks, it improves accuracy while reducing memory usage and training time, particularly for large 3D meshes.

  • Method: HI-MGN performs hierarchical message passing across multiple resolutions, using FPS, graph-based Voronoi partitioning, and topology-preserving coarse edges.This avoids proximity-based connections between disconnected components while preserving the original mesh topology.
  • Method: A learned geometry-aware GNN interpolator reconstructs fine-resolution representations from neighboring coarse states, fine-scale skip features, and relative geometric information.The interpolation retains fine-scale geometric information while enabling communication over long graph distances.
  • Results: Across all three benchmarks, HI-MGN consistently improves accuracy, while maintaining comparable or slightly improved accuracy for most steady aerodynamic quantities.The comparisons cover static structural, transient contact, and steady aerodynamic problems against MeshGraphNets and BSMS-GNN.
  • Results: Hierarchical message passing is most advantageous when predictions require large-distance information propagation, reducing errors caused by flat processors’ limited receptive fields.The benefit is especially pronounced for globally coupled thermoelastic fields and newly generated contact information shortly after contact onset.
  • Efficiency: 51.3% lower peak VRAM usage and 63.0% lower training time were achieved relative to the more computationally demanding baselines.HI-MGN remained competitive on 2D and surface meshes while reducing memory and training time significantly for 3D meshes.

Declaration of competing interest

The authors declare no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

  • The authors declare that they have no known competing financial interests.
  • The authors report no personal relationships that could have appeared to influence the work.
  • The declaration applies to the work reported in this paper.

Data availability

The HI-MGN source code and associated materials are publicly available, including core modules, training configurations, preprocessing scripts, and scripts for reproducing the main results.

  • The publicly available HI-MGN source code includes graph-coarsening and interpolation modules.
  • The release includes training configurations and benchmark preprocessing scripts.
  • Scripts used to reproduce the study’s main results are also publicly available.

Appendix A. Computational cost evaluation

The appendix evaluates computational cost using trainable parameters, preprocessing and inference times, peak GPU memory, and total training time. It distinguishes model complexity from implementation-dependent wall-clock cost and separately reports model-only latency for hierarchical models.

  • Evaluation metrics: Computational cost is assessed through trainable parameters, end-to-end preprocessing time, inference time, peak GPU memory, and total training time.These measures are reported alongside one another to distinguish model complexity from implementation-dependent wall-clock cost.
  • Hierarchical preprocessing: For hierarchical models, preprocessing includes constructing coarse graphs and inter-level mappings.
  • Latency measurement: Model-only latency is reported separately with all graph hierarchies precomputed.
  • Benchmark analyses: Additional computational information is provided for 2D static thermoelastic, 3D nonlinear contact, and 3D steady aerodynamic analyses.

Appendix B. Parameter matching BSMS-GNN

The appendix matches BSMS-GNN to HI-MGN by parameter count for 2D static thermoelastic analysis and finds that HI-MGN remains more accurate and efficient despite BSMS-GNN’s smaller parameter count.

  • Accuracy comparison: Matching the parameter count improves BSMS-GNN’s stress and displacement accuracy relative to its 128 latent dimension configuration.The appendix reports enhanced stress and displacement accuracy for the parameter-matched BSMS-GNN.
  • Accuracy comparison: HI-MGN still performs better than the 192-latent-dimension BSMS-GNN despite using fewer parameters.This comparison indicates that parameter count alone does not explain the accuracy difference between the models.
  • Efficiency comparison: BSMS-GNN’s VRAM usage and training time increase under parameter matching, while HI-MGN remains more efficient in both measures.The appendix specifically identifies peak memory usage and training time as efficiency advantages for HI-MGN.

Appendix C. Ablation study

The ablation study on 2D static thermoelastic analysis finds that HI-MGN accuracy is relatively insensitive to message-passing configuration but more dependent on its hierarchical representation and inter-level transfer operators. The results identify the hierarchical multi-scale structure as the primary source of accuracy, with processor configuration playing a secondary role.

  • Experimental setup: The ablation study varies one design axis at a time while keeping the encoder, decoder, training schedule, data split, and loss fixed.Experiments use 2D static thermoelastic analysis, where the largest accuracy variation is observed.
  • Message-passing configuration: Changing message-passing block distributions to flat or fine-centric configurations causes only moderate accuracy changes.The compared configurations are [6] and, versus the baseline [6] [8].
  • Hierarchical representation: Modifying the hierarchical representation produces larger accuracy losses than changing message-passing parameters.Three hierarchy stages reduce R2 to 0.9862, and direct FPS-seed latent inheritance yields an R2 of 0.9874.
  • Inter-level transfer: Replacing learned GNN interpolation with linear interpolation also decreases accuracy, showing sensitivity to inter-level transfer operators.The passage reports the direction of change but no numerical value for the linear-interpolation variant.
  • Overall finding: The hierarchical multi-scale structure is the primary accuracy source, while detailed processor configuration plays a secondary role after hierarchical communication is established.Performance remains relatively stable across substantial changes in the number and distribution of message-passing blocks, whereas hierarchy and transfer modifications cause more noticeable degradation.
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