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

cs.LGAug 13, 2026

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

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

Organizations: Mobile Experience, Digital Twin AX Lab., Samsung Electronics Co., 129, Samsung-ro, Yeongtong-gu, Suwon-si, 16677, Gyeonggi-do, Republic of Korea · Department of Aerospace system Engineering, Sejong University, 209, Neungdong-ro, Gwangjin-gu, Seoul, 05006, Republic of Korea

Abstract

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.

Figures & tables

Figure 1: Architecture of HI-MGN with hierarchical, multi-scale message passing (MP).

Figure 2: Coarsening and GNN interpolation in HI-MGN.

Figure 3: Schematics for 2D thermoelastic example

Property	Background ( $A$ )	Inclusion ( $B$ )
Elastic modulus	$1\ \mathrm{Pa}$	$1\ \mathrm{Pa}$
Poisson’s ratio	$0.3$	$0.3$
Thermal conductivity	$1\ \mathrm{W\,m^{-1}\,K^{-1}}$	$10\ \mathrm{W\,m^{-1}\,K^{-1}}$
Thermal expansion coefficient	$10^{-5}\ \mathrm{K^{-1}}$	$10^{-4}\ \mathrm{K^{-1}}$

Table 1: Material properties of the background and the inclusion.

Figure 4: Schematic and finite element discretization for the rectangular inclusion inference, with $a=b=2,000\,\mathrm{mm}$ and $c=d=450\,\mathrm{mm}$ .

Figure 5: Inferred results for geometric extrapolation with rectangular inclusion. All units are in Pa (stress) and mm (displacement).

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (28MP)
Stress $R^{2}$	0.99	0.91	0.85
Disp. $R^{2}$	0.99	0.94	-0.11

Table 2:

R^{2}

between the ground truth and rectangular inclusion inference results of HI-MGN, BSMS-GNN, and MeshGraphNets.

Metric	HI-MGN (proposed)	BSMS-GNN	MeshGraphNets
VRAM	10.45 GB	9.93 GB	17.03 GB
Train time	10.18 hr	8.46 hr	25.41 hr

Table 3: Training time and peak VRAM usage of HI-MGN, BSMS-GNN, and MeshGraphNets.

Figure 6: Schematic of the 3D contact example. All unspecified rounding fillet radii are $30\,\mathrm{mm}$ , and all units are in mm. The finite element discretization shown in isometric view corresponds to the example case of $a=240\,\mathrm{mm}$ and $\gamma_{a}=0.5$ .

Property	Indenter ( $A$ )	Foundation block ( $B$ )
Elastic modulus	$100\ \mathrm{Pa}$	$1\ \mathrm{Pa}$
Poisson’s ratio	$0.3$	$0.3$

Table 4: Material properties of the indenter and the foundation block.

Figure 7: Schematic and finite element discretization for the extrapolation sample of 3D contact problem, with $a=340\,\mathrm{mm}$ and $\gamma_{a}=0.05$ .

Figure 8: Inferred results for geometric extrapolation.

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (15MP)	MGN (28MP)
Stress $R^{2}$	0.90	0.83	0.72	0.77
Disp. $R^{2}$	0.99	0.98	0.96	0.97

Table 5: Time-averaged

R^{2}

between the ground truth and inference results of HI-MGN, BSMS-GNN, and MeshGraphNets.

Figure 9: $R^{2}$ results with respect to time.

* Jobs with VRAM exceeding 24GB are trained with two GPUs using pipeline parallelism.
Metric	HI-MGN (proposed)	BSMS-GNN	MGN (15MP)	MGN (28MP)
VRAM ∗	21.81 GB	23.12 GB	22.75 GB	42.16 GB
Train time	41.58 hr	112.61 hr	68.77 hr	90.47 hr

Table 6: Training time and peak VRAM usage of HI-MGN, BSMS-GNN, and MeshGraphNets.

Panel of Figure 10 — Figure 10: Inferred $C_{p}$ results for the first sample in the test set.

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (15MP)	MGN (28MP)
$C_{p}$ $R^{2}$	0.995	0.986	0.991	0.989
$C_{f_{x}}R^{2}$	0.968	0.964	0.966	0.967
$C_{f_{y}}R^{2}$	0.968	0.962	0.966	0.967
$C_{f_{z}}R^{2}$	0.995	0.994	0.995	0.996

Table 7: Sample-averaged

R^{2}

between the ground truth and inference results of HI-MGN, BSMS-GNN, and MeshGraphNets.

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (15MP)	MGN (28MP)
VRAM	7.68 GB	7.83 GB	9.80 GB	15.77 GB
Train time	12.32 hr	10.97 hr	16.42 hr	29.66 hr

Table 8: Training time and peak VRAM usage of HI-MGN, BSMS-GNN, and MeshGraphNets.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (28MP)
Number of parameters	4.667 M	2.218 M	4.267 M
Preprocessing time	234 s	144 s	77 s
Inference time	87.7 ms	330.6 ms	215.4 ms

Appendix

Table 9: Additional information regarding the 2D static thermoelastic analysis.

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (15MP)	MGN (28MP)
Number of parameters	5.493 M	2.218 M	3.852 M	7.126 M
Preprocessing overhead	978 s	1,071 s	477 s	477 s
Inference time	458.6 ms	2,183.8 ms	666.8 ms	1,230.0 ms

Appendix

Table 10: Additional information regarding the 3D nonlinear contact analysis.

Metric	HI-MGN (proposed)	BSMS-GNN	MGN (15MP)	MGN (28MP)
Number of parameters	4.668 M	2.880 M	2.334 M	4.267 M
Preprocessing overhead	720 s	360 s	117 s	117 s
Inference time	249.0 ms	460.2 ms	381.9 ms	698.9 ms

Appendix

Table 11: Additional information regarding the 3D steady aerodynamic analysis.

Metric	BSMS-GNN	HI-MGN
Latent dimension	192	128
Number of parameters	4.974 M	4.667 M
VRAM	14.83 GB	10.45 GB
Train time	11.26 hr	10.18 hr
Stress $R^{2}$	0.92	0.99
Disp. $R^{2}$	0.96	0.99

Appendix

Table 12: Numerical results of parameter matching BSMS-GNN on 2D static thermoelastic analysis.

Variant	$R^{2}$
Baseline	0.9949
1-stage hierarchy, [100]	0.9933
3-stage hierarchy, [10,000, 1,000, 100]	0.9862
Flat message passing: [5,6,6,6,5]	0.9918
Fine-centric message passing: [7,5,4,5,7]	0.9903
Number of MP blocks: 14	0.9920

Appendix

Table 13: Ablation study on 2D static thermoelastic analysis.

Metric	Value	Metric	Value
VRAM	24.02 GB	Train time	41.51 hr
$C_{p}$ $R^{2}$	0.9846	$C_{f_{x}}$ $R^{2}$	0.9801
$C_{f_{y}}$ $R^{2}$	0.9713	$C_{f_{z}}$ $R^{2}$	0.9900

Appendix

Table 14: Training and inference metrics of NASA-CRM model on HI-MGN.

Figure 11: Inferred $C_{p}$ results for a test sample with $M=0.877$ , $\alpha=3.457$ , $\phi_{outAil}=1.36$ , $\phi_{inAil}=-12.42$ , $\phi_{el}=-2.89$ , $\phi_{htp}=0.959$ .

Methods	$C_{p}$	$C_{f}$
Transolver-3	8.71%	5.85%
AB-UPT	9.77%	6.43%
HI-MGN (proposed)	11.31%	6.51%
Transolver++	9.51%	6.95%
Transolver	9.61%	7.04%
MGN (15MP)	14.80%	9.81%

Appendix

Table 15: Relative

L_{2}

error of NASA-CRM dataset on various methods [ 41 ] .

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