Flow Mesh: A Velocity-Regularized Flow Matching Framework for Stable Distributional Modeling of Physical Fields on Unstructured Meshes
Authors: Siddharth Verma, Jaewan Park, Jay Phil Yoo, Kazuma Kobayashi, Souvik Chakraborty, Syed Bahauddin Alam
Organizations: Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India · National Center for Supercomputing Applications, Urbana, IL, USA · Grainger College of Engineering, Mechanical Science & Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi
Thermal-fluid field distributions underpin the analysis and operation of energy infrastructure, from nuclear-reactor cooling channels to industrial heat exchangers, but resolving these distributions through computational fluid dynamics requires prohibitively large simulation ensembles. Their complex geometries also demand unstructured meshes that are poorly accommodated by generative models developed for regular grids. Here we introduce FlowMesh, a generative framework that combines permutation-invariant mesh encoding with deterministic latent flow matching and an l1 penalty on the learned transport velocity. FlowMesh generates full-field realizations directly on fixed unstructured meshes without auxiliary grid interpolation or a stochastic denoising chain. Across lid-driven-cavity, pressurized-water-reactor subchannel and heat-exchanger datasets, evaluated over six matched training seeds, FlowMesh achieves macro-averaged probability-density-function coverage of 91.7%-97.0%, exceeding an architecture-controlled latent diffusion model by 11.6--45.4 percentage points. Latent-velocity regularization increases coverage by 3.8 percentage points for reactor flows and 1.1 points for heat-exchanger flows, while reducing temperature-field Kullback--Leibler divergence by 1.7-2.4-fold. Each full-field realization requires 1.23 \pm 0.15 s, providing a 585--1902-fold wall-clock acceleration over computational fluid dynamics. FlowMesh therefore makes large-scale generative characterization of thermal-fluid states computationally feasible, providing a foundation for data-driven analysis and digital-twin development in fixed-geometry energy systems.
Figures & tables
Figure 1: FlowMesh formulates mesh-based field generation as deterministic latent transport. A permutation-invariant PointNet autoencoder maps physical fields and node coordinates from a fixed unstructured mesh into a compact latent representation. A learned time-dependent velocity field uθ(zt,t) then transports samples from a Gaussian source distribution p0 to the latent data distribution p1 through numerical integration of an ordinary differential equation. The decoder maps the terminal latent state back to the original mesh without interpolation to an auxiliary Cartesian grid. The schematic contrasts this continuous transport formulation with iterative denoising; it is conceptual and does not imply that individual learned trajectories coincide with a physical trajectory or an optimal-transport geodesic.
Figure 2: Lid-driven cavity benchmark results on an unstructured mesh. (a) Two-dimensional lid-driven cavity configuration with prescribed boundary conditions and the turbulent kinetic energy (TKE) field at the terminal simulation time. (b) Probability density function (PDF) comparisons between the reference computational fluid dynamics (CFD) data and FlowMesh-generated samples for pressure, velocity magnitude, and TKE. (c) Node-wise diagnostics for velocity magnitude, including mean and variance fields from the reference data and generated samples with and without regularization, mean-error maps relative to the reference, and Kullback-Leibler (KL) and Jensen-Shannon (JS) divergence maps between the reference and generated node-wise distributions.
λ
Pressure
Velocity
TKE
0
96.0±1.6
95.7±1.0
97.7±1.3
10−4
96.2±2.0
95.4±0.9
97.7±1.4
10−3
96.2±1.2
95.7±0.8
97.8±1.2
10−2
96.4±1.4
95.7±0.8
97.6±1.2
10−1
96.8±0.9
96.0±0.7
98.1±0.8
1.0
50.8±1.5
14.9±0.1
60.1±3.4
Table 1: LDC ℓ1 velocity-regularization sweep across 6 independent training seeds (PDF coverage, mean ± standard deviation, evaluated against autoencoder-reconstructed reference fields). Coverage varies by less than approximately one percentage point for 0≤λ≤10−1 , whereas λ=1 produces a large loss of fidelity in all three fields. The result establishes a broad low-to-moderate regularization regime in which the cavity solution is insensitive to λ , followed by clear over-regularization at the largest weight.
Model
Field
KL ↓
JS ↓
PDF Coverage (%) ↑
FlowMesh ℓ1 No Reg.
Pressure
0.0071±0.0058
0.0027±0.0024
96.0±1.6
Velocity
0.0173±0.0063
0.0054±0.0017
95.7±1.0
TKE
0.0027±0.0019
0.0010±0.0007
97.7±1.3
FlowMesh ℓ1 , Reg. ( λ=10−3 )
Pressure
0.0060±0.0041
0.0023±0.0016
96.2±1.2
Velocity
0.0180±0.0071
0.0057±0.0020
95.7±0.8
TKE
0.0023±0.0018
0.0008±0.0006
97.8±1.2
Table 2: LDC distributional metrics over 6 training seeds (mean ± standard deviation). All configurations share the same architecture, data, and 100-epoch training schedule. Coverage is evaluated against autoencoder-reconstructed reference fields. Regularization changes coverage by less than one percentage point, within the seed-to-seed variability. Lower KL and JS and higher PDF coverage are better.
Data Origin
Field
Mean
Min Value
Max Value
Original
Pressure
−234.7000
−1570.1500
2515.7000
Velocity
0.0320
0.0000
0.7470
TKE
0.3130
0.0000
7.4450
FlowMesh No Regularizer
Pressure
−226.9062
−1239.5104
2140.1989
Velocity
0.0206
−0.0051
0.7677
TKE
0.3086
−0.1371
6.9881
Table 3: LDC (lid-driven cavity): comparison of statistical values of data across various fields. FlowMesh rows are the mean over 6 independent training seeds (same architecture, data and 100-epoch schedule; only the training seed differs); the Original row is as reported for the original dataset. Negative velocity-magnitude & TKE values are physically inadmissible; the table reports raw model outputs prior to any non-negativity post-processing.
Figure 3: Pressurized water reactor (PWR) subchannel benchmark results on an unstructured mesh. (a) Schematic of the reactor subchannel geometry showing the fuel rods, coolant inlet conditions, flow direction, and extracted transverse cross-section, illustrated using the temperature field. (b) Probability density function (PDF) comparisons between the reference computational fluid dynamics (CFD) data and FlowMesh-generated samples for velocity magnitude, temperature, and turbulent kinetic energy (TKE), with and without regularization. (c) Node-wise diagnostics for temperature, including mean and variance fields from the reference data and generated samples with and without regularization, mean-error maps relative to the reference, and Kullback-Leibler (KL) and Jensen-Shannon (JS) divergence maps between the reference and generated node-wise distributions.
Model
Field
KL ↓
JS ↓
PDF Coverage (%) ↑
FlowMesh ℓ1 No Reg.
Velocity
0.0613±0.0454
0.0221±0.0169
89.6±5.0
Temperature
0.1517±0.1041
0.0545±0.0339
81.6±6.6
TKE
0.0350±0.0323
0.0118±0.0105
92.5±3.8
FlowMesh ℓ1 , Reg. ( λ=10−3 )
Velocity
0.0323±0.0186
0.0114±0.0055
92.5±2.5
Temperature
0.1350±0.1675
0.0536±0.0657
83.7±12.1
TKE
0.0142±0.0095
0.0050±0.0031
95.1±1.9
Table 4: PWR sub-channel distributional metrics over six fixed, matched training seeds (mean ± standard deviation). All configurations share the same architecture, data, and 100-epoch training schedule. FlowMesh is evaluated against autoencoder-reconstructed reference fields. Lower KL and JS and higher PDF coverage are better.
Data Origin
Field
Mean
Min Value
Max Value
Original
Velocity
3.992
0.000
5.250
Temperature
573.880
512.260
649.680
TKE
0.056
0.011
0.146
FlowMesh, No Reg.
Velocity
4.024
−0.263
5.461
Temperature
575.795
503.519
659.990
TKE
0.057
0.009
0.156
Table 5: PWR sub-channel field statistics. Each FlowMesh entry is the average of the corresponding statistic over the 6 training seeds. The Original row reports statistics from the reference dataset. Negative velocity-magnitude & TKE values are physically inadmissible; the table reports raw model outputs prior to any non-negativity post-processing.
Model
Field
KL ↓
JS ↓
Wasserstein ↓
PDF Coverage (%) ↑
Flow Match – No Reg ( ℓ1 , λ=0 )
Velocity
0.0276
0.0112
0.0650
92.93
Temperature
0.0322
0.0137
1.8165
92.77
TKE
0.0117
0.0044
0.0016
95.27
Flow Match – With Reg ( ℓ1 , λ⋆=10−3 )
Velocity
0.0175
0.0075
0.0255
94.40
Temperature
0.0310
0.0134
2.6921
93.07
TKE
0.0047
0.0018
0.0006
96.98
Table 6: PWR sub-channel single-run comparison of ℓ1 and ℓ2 flow-matching supervision and the autoencoder reconstruction ceiling. For each loss, the unregularized baseline ( λ=0 ) is compared with the selected regularized setting ( λ=10−3 ). The autoencoder reconstruction diagnostic passes N=4,000 reference fields through Dψ∘Eϕ and evaluates them using the same metrics; it upper-bounds the distributional fidelity attainable by the latent flow model with this autoencoder. Lower KL, JS, and Wasserstein distance and higher PDF coverage are better.
Figure 4: Heat exchanger benchmark results on an unstructured mesh. (a) Schematic of the heat exchanger configuration showing the flow direction, internal geometry, and transverse cross-section used for virtual sensing, with velocity magnitude visualized on the sensor plane. (b) Probability density function (PDF) comparisons between the reference computational fluid dynamics (CFD) data and FlowMesh-generated samples with and without regularization for turbulent kinetic energy (TKE), pressure, temperature, three velocity components, and velocity magnitude. (c) Node-wise diagnostics for velocity magnitude, including mean and variance fields from the reference data and generated samples with and without regularization, mean-difference maps relative to the reference, and Kullback–Leibler (KL) and Jensen–Shannon (JS) divergence maps between the reference and generated node-wise distributions.
Model
Field
KL ↓
JS ↓
PDF Coverage (%) ↑
FlowMesh No Regularizer
TKE
0.00382
0.00142
97.800
Pressure
0.00489
0.00177
96.162
Temperature
0.18662
0.06622
80.288
Z-Velocity
0.00038
0.00014
99.196
Y-Velocity
0.00189
0.00070
98.586
X-Velocity
0.00339
0.00134
97.784
Table 7: Comparison of FlowMesh metrics (KL, JS, PDF coverage) on the Heat-Exchanger dataset, averaged over 6 independent training seeds . Coverage is evaluated against the autoencoder-reconstructed reference. Lower is better for KL and JS.
Data Origin
Field
Mean
Min Value
Max Value
Original
TKE
1.0780
0.0490
2.9885
Pressure
6645.9000
−13974.3800
25704.1000
Temperature
292.3600
263.8500
322.5100
Z-Velocity
0.0009
−4.4910
4.5110
Y-Velocity
0.1602
−2.1800
3.4880
X-Velocity
3.9170
0.0000
9.2520
Table 8: Comparison of statistical values of data across various fields (Heat-Exchanger). FlowMesh rows are the mean over 6 independent training seeds ; the Original row is as reported for the original dataset.
Figure 5: Conceptual interpretation of latent-velocity regularization. (a) Unregularized flow matching is free to use latent velocities of any magnitude required by the data-matching objective. Large or rapidly varying transport fields can increase sensitivity of the numerical trajectory to local approximation error. The illustration is schematic and should not be interpreted as a measured trajectory or direct evidence of “manifold detachment.” (b) FlowMesh augments the flow-matching loss with an ℓ1 penalty on the latent velocity magnitude. The regularization sweep in Fig. 6 shows that increasing λ is empirically associated with a reduction in the sampled Jacobian norm of the learned velocity field, while moderate values can improve distributional fidelity. The implemented penalty is not a Dirichlet-energy objective and does not establish Wasserstein-geodesic optimality or formal contractivity.
Dataset
FlowMesh (Ours)
Flow Matching (Lipman et al.)
Latent DDPM
Lid-driven Cavity
96.97
96.47
85.39
PWR Subchannel
91.67
87.90
46.30
Heat Exchanger
96.45
95.39
82.70
Table 9: Comparison of PDF coverage (%) between FlowMesh, vanilla flow matching (Lipman et al.), and a latent DDPM baseline across benchmark datasets. FlowMesh and vanilla flow matching values are unweighted macro-averages across all reported fields using six fixed, matched training seeds. FlowMesh uses ℓ1 velocity regularization with λ=10−1 , whereas vanilla flow matching uses the identical model with λ=0 . All three models share the same PointNet autoencoder and latent space. Higher values indicate better distributional fidelity.
Metric
FlowMesh
Latent DDPM (shared PointNet encoder)
Network parameters
20.8 M
6.1 M
Forward pass
15.1 GFLOP
10.6 GFLOP
Training wall-clock (100 epochs)
7.6 min
39.5 min
Peak training memory
9.4 GB
6.1 GB
NFE per sample
94.7±11.6 (adaptive)
250 (DDIM)
Inference cost per sample
∼ 1.4 TFLOP
∼ 2.6 TFLOP
Table 10: Computational cost of FlowMesh versus the latent DDPM baseline. Both models operate on identical PointNet latent spaces; reported differences reflect only the generative component. Measured on one NVIDIA A100 (40 GB) GPU on the 4,096-dimensional latent (reshaped to 1×64×64 ), which is identical across datasets; training times were measured on the PWR subchannel latents, with FlowMesh at λ=0 ; FLOPs from the PyTorch operator-level profiler. Training uses a matched 100-epoch budget and a batch size of 128; FlowMesh trains in mixed precision (bf16), and DDPM in fp32, following each reference implementation.
Dataset
Mesh Nodes
CFD (s/sample)
FlowMesh (s/sample)
Speed-up †
Lid-driven Cavity (LDC)
4225
2340
1.23 0.15
1902
PWR Subchannel
1733
720
1.23 0.15
585
Heat Exchanger (HX)
3977
1800
1.23 0.15
1463
Table 11: Per-sample computation time for each benchmark dataset.CFD reference times correspond to single-case ANSYS Fluent runs on a 12-core Xeon CPU. FlowMesh inference is measured on a single NVIDIA A100 (40 GB) GPU (batch size 1, adaptive Dormand–Prince ODE solver, regularised model λ=10−3 ), averaged over 1,000 samples following 20 warm-up runs.Speed-up is the ratio of the CFD wall-clock time to the FlowMesh wall-clock time.
Figure 6: Regularization sweep maps the accuracy–stability trade-off in latent transport. PWR subchannel benchmark, single-run sweep. (a) Temperature KL divergence as a function of regularization strength λ∈{0,10−4,10−3,10−2,10−1,1} . Under ℓ1 supervision, KL decreases from the unregularized value of approximately 0.032 to approximately 0.022 at λ=10−4 and remains comparatively low through moderate λ , before increasing sharply at λ=1 . The ℓ2 model deteriorates at smaller regularization weights. (b) Empirical Jacobian-norm proxy J^=max(z,t)∈S∥∇zuθ(z,t)∥F estimated on the sampled evaluation set S . The proxy decreases from approximately 65 at small λ to approximately 28 for ℓ1 and 32 for ℓ2 at λ=1 . This is an empirical measure of local velocity-field variation, not a proof of a global Lipschitz bound. (c) Temperature and velocity PDF coverage versus λ . Small-to-moderate ℓ1 regularization improves both fields in this single run, whereas excessive regularization causes both to collapse. Together, the panels show that useful regularization occurs in an intermediate regime: reducing latent-velocity freedom can improve distributional transport, but suppressing it too strongly prevents the model from reaching the target distribution.
Computational fluid dynamics (CFD) provides high-fidelity simulations of fluid flows but remains computationally expensive for many-query applications. In recent years deep learning (DL) has been used to construct data-driven fluid-dynamic surrogate models. In this work we consider a different learning paradigm and embrace generative modelling as a framework for constructing scalable fluid-dynamics surrogate models. We introduce FluidFlow, a generative model based on conditional flow-matching, a recent alternative to diffusion models that learns deterministic transport maps between noise and data distributions. FluidFlow is specifically designed to operate directly on CFD data defined on both structured and unstructured meshes alike, without the needs to perform any mesh interpolation pre-processing and preserving geometric fidelity. We assess the capabilities of FluidFlow using two different core neural network architectures, a U-Net and diffusion transformer (DiT), and condition their learning on physically meaningful parameters. The methodology is validated on two benchmark problems of increasing complexity: prediction of pressure coefficients along an airfoil boundary across different operating conditions, and prediction of pressure and friction coefficients over a full three-dimensional aircraft geometry discretized on a large unstructured mesh. In both cases, FluidFlow outperform strong multilayer perceptron baselines, achieving significantly lower error metrics and improved generalisation across operating conditions. Notably, the transformer-based architecture enables scalable learning on large unstructured datasets while maintaining high predictive accuracy. These results demonstrate that flow-matching generative models provide an effective and flexible framework for surrogate modelling in fluid dynamics, with potential for realistic engineering and scientific applications.
David Ramos, Lucas Lacasa, Fermín Gutiérrez +2
ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, Spain · Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, CSIC-UIB), 07122 Palma de Mallorca (Spain) · Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain
Meshes are among the most common 3D scene representations, but directly generating meshes is challenging because the representation contains important symmetries, including permutation invariance of faces and vertices. MeshFlow learns to generate triangle meshes directly as triangle soups, avoiding the need to serialize meshes into long autoregressive sequences. We adopt equivariant optimal-transport flow matching models that respect the key symmetries of triangle soups: arbitrary permutations of faces and permutations of the vertices within each face. Toward this goal, we propose a simple yet effective modification to the Diffusion Transformer architecture, resulting in a scalable network capable of modeling a velocity field while maintaining the desired equivariance. We further introduce an optimal-transport-based training objective that improves convergence by eliminating supervision signals that violate these symmetries. MeshFlow achieves mesh quality comparable to state-of-the-art autoregressive mesh generators while providing about an 18× speedup during inference. Project page is at https://qiisun.github.io/MeshFlow/.
Qi Sun, Kiyohiro Nakayama, Jing Nathan Yan +6
City University of Hong Kong, Hong Kong · Stanford University, USA · Cornell Tech, USA +1
Analyzing unsteady fluid flows often requires access to the full distribution of possible temporal states, yet conventional PDE solvers are computationally prohibitive and learned time-stepping surrogates quickly accumulate error over long rollouts. Generative models avoid compounding error by sampling states independently, but diffusion and flow-matching methods, while accurate, are limited by the cost of many evaluations over the entire mesh. We introduce scale-autoregressive modeling (SAR) for sampling flows on unstructured meshes hierarchically from coarse to fine: it first generates a low-resolution field, then refines it by progressively sampling higher resolutions conditioned on coarser predictions. This coarse-to-fine factorization improves efficiency by concentrating computation at coarser scales, where uncertainty is greatest, while requiring fewer steps at finer scales. Across unsteady-flow benchmarks of varying complexity, SAR attains substantially lower distributional error and higher per-sample accuracy than state-of-the-art diffusion models based on multi-scale GNNs, while matching or surpassing a flow-matching Transolver (a linear-time transformer) yet running 2-7x faster than this depending on the task. Overall, SAR provides a practical tool for fast and accurate estimation of statistical flow quantities (e.g., turbulent kinetic energy and two-point correlations) in real-world settings.