Predicting Sparse Observables in Parameterized Unsteady Flows Using Coarse-Mesh Graph Neural Networks

Brener Ramos; William R. Wolf; Tarcísio C. D. Oliveira; Scott T. M. Dawson

cs.LGOct 27, 2026

Predicting Sparse Observables in Parameterized Unsteady Flows Using Coarse-Mesh Graph Neural Networks

Authors: Brener Ramos, William R. Wolf, Tarcísio C. D. Oliveira, Scott T. M. Dawson

Organizations: School of Mechanical Engineering, University of Campinas, Campinas, São Paulo, Brazil · Department of Mechanical, Materials, and Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois, USA

Abstract

Full-order surrogates for high-dimensional nonlinear flows are often unnecessary when applications require only selected degrees of freedom. We propose reduced-order models that bypass full-state reconstruction by deploying graph neural networks (GNNs) directly on coarse meshes. Applied to parameterized 2D/3D confined cylinder flows and an unevenly heated cavity over Reynolds/Rayleigh-number ranges, the models generalize to unseen conditions and accurately predict sparse probe locations and integrated quantities. Coarse-mesh GNNs match or outperform finer-mesh surrogates while reducing training time and memory, even for 3D chaotic dynamics. A dynamics-informed adaptive sampling strategy guided by frequency-spectrum variations reduces maximum errors by an order of magnitude versus uniform sampling at negligible cost. Coarse-mesh training also makes noise-augmented multi-step unrolling practical, improving long-horizon autoregressive predictions. The approach outperforms parametric DMD, sparse MLPs, U-Nets, and Fourier neural operators at lower cost, and enables system identification and state estimation from sparse observations. These results show that coarse-mesh representations, adaptive sampling, and noise-augmented unrolled training form an efficient framework for reduced-order modeling of parameterized unsteady flows without solver-resolution reconstruction.

Figures & tables

Figure 1: Architecture of GNNs used in this work. Red and blue arrows denote variables in edge (matrices with $E$ rows) and node (matrices with $N$ rows) spaces.

Figure 2: Contours of vorticity for the 2D confined cylinder flow at $\mbox{{Re}}=100$ (top) and $\mbox{{Re}}=1000$ (bottom). Contours in white show attracting Lagrangian coherent structures through the ridges of backward finite time Lyapunov exponents. Location of probes and integration regions used for evaluating models are highlighted in pink.

Figure 3: Contours of vorticity for the cavity flow at $\mbox{{Ra}}=4\times 10^{5}$ (left) and $\mbox{{Ra}}=6\times 10^{5}$ (right). For each Ra , the top and bottom portions of the cavity are shown separately for better visualization. Contours in white show attracting Lagrangian coherent structures through the ridges of backward finite time Lyapunov exponents. Location of probes and integration regions used for evaluating models are highlighted in pink.

Figure 4: Q-criterion isosurfaces with contours of vorticity for the 3D confined cylinder system at $\mbox{{Re}}=100$ (top) and $\mbox{{Re}}=200$ (bottom). Location of probes used for evaluating models are highlighted in pink. The whole cylinder surface is the integration surface for this configuration.

Panel of Figure 5 — Figure 5: Simulation domains for the confined cylinder (left) and cavity (right) systems, respectively, where quadrilaterals represent spectral elements.

Panel of Figure 6 — Figure 6: 2D cylinder dynamics at different Re .

Panel of Figure 7 — Figure 7: Cavity flow dynamics according to Ra .

Panel of Figure 8 — Figure 8: 3D cylinder dynamics at different Re . Horizontal (left) and and spanwise (right) velocity signals taken at $x=3.2,y=-0.05$ .

System	Name	# Nodes	# Edges
Cylinder (2D)	C1	108	240
Cylinder (2D)	C2	239	611
Cylinder (2D)	C3	461	1259
Cylinder (2D)	C4	978	2782
Cylinder (2D)	C5	2083	6061
Cavity	K1	322	755

Table 1: Unstructured mesh attributes.

Figure 9: Adaptive parameter sampling procedure.

Figure 10: Iterative addition of parameter samples using the proposed adaptive sampling algorithm.

Figure 11: Schematic of unrolling procedure with two timesteps and assuming a node with only a one-dimensional feature.

Panel of Figure 13 — Figure 13: Cylinder (top) and cavity (bottom) surrogate errors by mesh size. Dashed lines represent maximum values of $e_{P}$ . Filled and empty bars of $e_{P}$ represent tests with parameters present and not present in the training dataset, respectively.

Panel of Figure 14 — Figure 14: Effect of the number of message-passing blocks on $\overline{e_{P}}$ for the cylinder (left) and cavity (right) flows. All models have approximately the same number of trainable parameters.

Figure 15: Receptive field sizes for Mesh C3 (left) and Mesh C5 (right). Contours represent $u_{x}$ fluctuations at $\mbox{{Re}}=1000$ .

Figure 16: Receptive field sizes for Mesh K2 (left) and Mesh K5 (right). Contours represent $u_{x}$ fluctuations at $\mbox{{Ra}}=600k$ .

Panel of Figure 17 — Figure 17: Cylinder (top) and cavity (bottom) surrogate model errors for different training datasets. Dashed lines represent maximum values of $e_{P}$ . Filled and empty bars of $e_{P}$ represent tests with parameters present and not present in the training dataset, respectively.

Panel of Figure 18 — Figure 18: Errors for the cylinder (top) and cavity (bottom) cases for different unrolling horizons. Bars represent maximum and minimum values of $e_{P}$ . Filled bars correspond to test parameters seen during training but with different initial conditions.

Figure 19: Training time as a function of unrolling horizon. Values are normalized by the time for $R=1\text{(K2)}$ .

Figure 20: Probe signals at $(x,y)=(0.6,-0.05)$ for cylinder flow at various Reynolds numbers. Phase shifts are significantly reduced by the proposed approach incorporating both mesh-coarsening and unrolling.

Figure 21: Probe signals at $(x,y)=(0.25,7.59)$ for cavity flow at various Rayleigh numbers. The proposed method maintains accuracy even as the flow transitions from periodic to intermittent behavior.

Figure 22: Drag coefficient $(C_{D})$ comparison for cylinder flow.

Figure 23: Nusselt number ( Nu ) comparison for cavity flow. The proposed method K2 (R8) captures the transition from periodic to intermittent dynamics more accurately than the baseline setup K5 (R1).

Panel of Figure 24 — Figure 24: Errors for the cylinder (top) and cavity (bottom) cases for different types of models. Bars represent maximum and minimum values of $e_{P}$ . Filled bars correspond to test parameters seen during training but with different initial conditions.

Panel of Figure 25 — Figure 25: Estimation of Re (left) and Ra (right) from various sparse reference signals across the parameter space.

Panel of Figure 26 — Figure 26: Estimation of $p$ and $u_{y}$ for the cylinder (left) and cavity (right) cases, respectively. Reference signals are acquired at $\mbox{{Re}}=900$ and $\mbox{{Ra}}=6\times 10^{5}$ , respectively.

Figure 27: Errors for the 3D cylinder case for different mesh refinement levels and models. Bars represent maximum and minimum values of $e_{P}$ . Filled bars correspond to test parameters seen during training but with different initial conditions.

Figure 28: $u_{z}$ signals at $(x,y)=(3.2,-0.05)$ for 3D cylinder flow at various Re numbers.

Figure 29: $u_{x}$ signals at $(x,y)=(3.2,-0.05)$ for 3D cylinder flow at various Re numbers.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 30 — Figure 30: Effect of the polynomial order on the drag coefficient across different Re (left) and on the Nusselt number across different Ra (right). Markers represent the mean values while the bars represent maximum and minimum values along the limit cycle. For $\mbox{{Ra}}=600k$ , results are shown on the right y-axis (red).

Figure 31: Meshes employed to trained models for the cylinder flow case.

Figure 32: Meshes employed to trained models for the cavity flow case. Only half of the domain is being shown.

Figure 33: Meshes employed to trained models for the 3D cylinder flow case.

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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.

Abstract

PDE Surrogate Modeling

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