Super-resolution of turbulent reacting flows on complex meshes using graph neural networks

Priyabrat Dash; Konduri Aditya; Christos E. Frouzakis; Mathis Bode

physics.flu-dynMar 1, 2026

Super-resolution of turbulent reacting flows on complex meshes using graph neural networks

Authors: Priyabrat Dash, Konduri Aditya, Christos E. Frouzakis, Mathis Bode

Organizations: FLAME Laboratory, Department of Computational and Data Sciences, Indian Institute of Science, Bengaluru, 560012, India · CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, 8092 Zürich, Switzerland · Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH, Jülich, 52425, Germany

Abstract

State-of-the-art deep learning models have been extensively utilized to reconstruct small-scale structures from coarse-grained data in turbulent flows. However, their application has predominantly been restricted to structured uniform meshes, limiting their applicability to data associated with complex geometries that are typically simulated on structured non-uniform or unstructured meshes. Machine learning (ML) models based on graph neural networks (GNNs), known for their ability to process unstructured data, offer a promising alternative. In this study, we leverage the inherent flexibility of GNNs featuring message passing layers to develop a methodology for reconstructing unresolved small-scale structures from low-resolution data on complex meshes. The accuracy of the proposed approach is demonstrated using two cases: a reacting channel flow on a structured non-uniform mesh, and a reacting hydrogen fueled internal combustion (IC) engine featuring an unstructured mesh. Evaluation of results based on visual agreement, statistical metrics, and cumulative error reduction indicates the effectiveness of the method in accurately reconstructing fine-scale features. Overall, this study provides a pathway for integrating data-driven small-scale reconstruction and subgrid-scale modeling to enhance the accuracy of coarse-grained simulations on complex meshes.

Figures & tables

Fig. 1: A schematic of the GNN framework for super-resolution (left) , and message passing layer (right) showing edge feature update, edge feature aggregation, and node feature update.

	Case 1	Case 2
Mesh	Structured non-uniform	Unstructured
Geometry	Canonical	Real-scale
Solver	Compressible	Low Mach
Nodes/instance	63.06 million	3.78 billion
DoFs/node	29	14
Demonstration	Comparison with CNN baseline	OOD extrapolation.

Table 1: Table outlining differences between Cases 1 and 2.

Fig. 2: Volumetric rendering of temperature inside the channel, with the flame surface represented in black color.

Fig. 3: Flame front isosurface superimposed on the normalized velocity magnitude distribution on the engine midplane.

Panel of Figure 5 — Fig. 4: Error reduction by GNN compared to trilinear interpolation and CNN baseline for Case 1. Column 1: Contours of key variables shown for the center-plane along $z$ . Columns 2, 3, and 4: Contours of absolute errors $\epsilon$ between ground truth (DNS solution) and the results of interpolation, CNN, and GNN.

Fig. 5: Cumulative error percentage for selected scalars across all upsampling methods in Case 1.

Region	Model	$u_{1}$	$u_{2}$	$u_{3}$	$T$	$\rho$	$Y_{CH_{4}}$	$Y_{O_{2}}$	$Y_{O}$	$Y_{OH}$	$Y_{CH_{2}O}$	$Y_{CO}$	$Y_{CO_{2}}$	$Y_{H_{2}O}$
Bottom wall	GNN	1.17	9.02	5.22	0.58	0.93	0.47	0.25	2.90	2.80	0.77	0.72	0.07	0.11
	Interpolation	1.76	13.85	8.39	1.17	2.59	0.78	0.41	4.34	4.72	1.07	1.17	0.08	0.14
	CNN	6.22	14.67	11.44	3.44	8.22	1.31	0.70	18.82	16.70	1.98	2.20	0.28	0.31
Center	GNN	0.23	3.01	2.90	0.07	0.12	0.29	0.17	0.36	0.30	0.35	0.22	0.05	0.04
	Interpolation	0.22	5.78	5.20	0.15	0.19	1.02	0.59	1.17	1.04	1.23	0.69	0.13	0.11
	CNN	0.51	3.07	3.08	0.13	0.79	0.73	0.39	1.42	1.49	0.98	0.51	0.08	0.10

Table 2: Percentage error across regions for each scalar across different upsampling approaches.

Panel of Figure 8 — Fig. 6: Joint PDFs computed between interpolation/network output and ground truth for transverse component of velocity and mass fraction of three species for Case 1. Grayscale color scheme is used for interpolation and jet colormap for the GNN output.

Fig. 7: Comparison of the root mean squared profiles of scalar gradients plotted along the transverse direction (

y

). Row 1: gradients in the streamwise direction (

x

) and Rows 2,3: gradients in the transverse direction (

y

Fig. 8: Error reduction by GNN compared to interpolation for Case 2. Column 1: Contours of key scalars on a slice approximately dividing the flame front into two equal halves along the $z$ direction. Deviation from ground truth is presented as absolute error for interpolation ( Column 2 ) and GNN output ( Column 3 ) computed for temperature, mass fractions of $\text{H}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}$ , OH , $\text{HO}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}$ , and $\text{H}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{O}$ , and $y-$ velocity.

Panel of Figure 11 — Fig. 9: Joint PDFs computed between interpolation/network output and ground truth for temperature and mass fractions of three species for Case 2. Grayscale color scheme is used for interpolation and the jet colormap is used for GNN output.

	$u_{1}$	$u_{2}$	$u_{3}$	$p$	$T$	$Y_{{}\mathrm{H}{\vphantom{\mathrm{X}}}_{\smash[t]{\mathrm{2}}}}$	$Y_{{}\mathrm{O}{\vphantom{\mathrm{X}}}_{\smash[t]{\mathrm{2}}}}$	$Y_{{}{}\mathrm{H}{\vphantom{\mathrm{X}}}_{\smash[t]{\mathrm{2}}}\mathrm{O}}$	$Y_{{}\mathrm{H}}$	$Y_{{}\mathrm{O}}$	$Y_{{}\mathrm{OH}}$	$Y_{{}\mathrm{HO}{\vphantom{\mathrm{X}}}_{\smash[t]{\mathrm{2}}}}$	$Y_{{}{}\mathrm{H}{\vphantom{\mathrm{X}}}_{\smash[t]{\mathrm{2}}}\mathrm{O}{\vphantom{\mathrm{X}}}_{\smash[t]{\mathrm{2}}}}$
Interpolation	5.23	5.30	6.63	0.75	0.67	0.23	0.13	1.22	31.46	16.38	4.66	42.27	27.65
Super-resolution	4.38	4.46	5.35	0.62	0.54	0.19	0.10	0.96	25.36	12.88	3.64	34.27	22.44
Reduction (%)	16.25	15.85	19.31	17.33	19.40	17.39	23.08	21.31	19.39	21.37	21.89	18.93	18.84

Table 3: Comparison of percentage errors (%) for IDW interpolation and super-resolution across all scalars.

Fig. 10: Normalized loss curves for reacting channel (Case 1, top) and reacting IC engine (Case 2, bottom).

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A multi-scale feature enhanced graph neural network for fluid dynamics prediction in complex geometries

Li Xiao, Tianyu Li, Yiye Zou, Mingjie Zhang+1

Graph Neural Networks PDE Surrogate Modeling Multi-Scale Feature Fusion Computational Fluid Dynamics

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Industrial design in fields such as vehicle and aerospace engineering often relies on large-scale numerical simulations to evaluate fluid dynamics performance, which can incur substantial computational costs. Deep neural networks have shown promise in improving simulation efficiency, especially graph neural networks (GNNs), which demonstrate great potential due to their flexibility with unstructured data. However, GNNs face challenges when dealing with tasks involving complex geometries and large-scale meshes. In this paper, we propose the Multi-scale Feature Enhanced Graph Neural Network (ME-GNN) to tackle these challenges. ME-GNN employs a graph neural network with a two-step message-passing mechanism to capture detailed local features effectively. Additionally, it integrates an Attention U-Net with uniform grid discretization, enabling the extraction of both fine and coarse features. The model also utilizes K-hop sampling to construct subgraphs, facilitating efficient training on large datasets while preserving detailed local features. We evaluated ME-GNN on three benchmark datasets and achieved state-of-the-art results: a relative L2 error of 0.0196 for the velocity field and 0.0556 for the surface pressure on ShapeNet-Car, a normalized mean squared error of 0.0033 for the flow field on AirfRANS, and a relative L2 error of 0.1416 for the surface pressure on DrivAerNet.

Abstract

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