Flow Mesh: A Velocity-Regularized Flow Matching Framework for Stable Distributional Modeling of Physical Fields on Unstructured Meshes

Siddharth Verma; Jaewan Park; Jay Phil Yoo; Kazuma Kobayashi; Souvik Chakraborty; Syed Bahauddin Alam

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

\mathbf{u}_{\theta}(\mathbf{z}_{t},t)

then transports samples from a Gaussian source distribution

p_{0}

to the latent data distribution

p_{1}

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.

$\lambda$	Pressure	Velocity	TKE
$0$	$96.0\pm 1.6$	$95.7\pm 1.0$	$97.7\pm 1.3$
$10^{-4}$	$96.2\pm 2.0$	$95.4\pm 0.9$	$97.7\pm 1.4$
$10^{-3}$	$96.2\pm 1.2$	$95.7\pm 0.8$	$97.8\pm 1.2$
$10^{-2}$	$96.4\pm 1.4$	$95.7\pm 0.8$	$97.6\pm 1.2$
$10^{-1}$	$96.8\pm 0.9$	$96.0\pm 0.7$	$98.1\pm 0.8$
$1.0$	$50.8\pm 1.5$	$14.9\pm 0.1$	$60.1\pm 3.4$

Table 1: LDC

\ell_{1}

velocity-regularization sweep across 6 independent training seeds (PDF coverage, mean

\pm

standard deviation, evaluated against autoencoder-reconstructed reference fields). Coverage varies by less than approximately one percentage point for

0\leq\lambda\leq 10^{-1}

, whereas

\lambda=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

\lambda

, followed by clear over-regularization at the largest weight.

Model	Field	KL $\downarrow$	JS $\downarrow$	PDF Coverage (%) $\uparrow$
FlowMesh $\ell_{1}$ No Reg.	Pressure	$0.0071\pm 0.0058$	$0.0027\pm 0.0024$	$96.0\pm 1.6$
	Velocity	$0.0173\pm 0.0063$	$0.0054\pm 0.0017$	$95.7\pm 1.0$
	TKE	$0.0027\pm 0.0019$	$0.0010\pm 0.0007$	$97.7\pm 1.3$
FlowMesh $\ell_{1}$ , Reg. ( $\lambda=10^{-3}$ )	Pressure	$0.0060\pm 0.0041$	$0.0023\pm 0.0016$	$96.2\pm 1.2$
	Velocity	$0.0180\pm 0.0071$	$0.0057\pm 0.0020$	$95.7\pm 0.8$
	TKE	$0.0023\pm 0.0018$	$0.0008\pm 0.0006$	$97.8\pm 1.2$

Table 2: LDC distributional metrics over 6 training seeds (mean

\pm

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 $\downarrow$	JS $\downarrow$	PDF Coverage (%) $\uparrow$
FlowMesh $\ell_{1}$ No Reg.	Velocity	$0.0613\pm 0.0454$	$0.0221\pm 0.0169$	$89.6\pm 5.0$
	Temperature	$0.1517\pm 0.1041$	$0.0545\pm 0.0339$	$81.6\pm 6.6$
	TKE	$0.0350\pm 0.0323$	$0.0118\pm 0.0105$	$92.5\pm 3.8$
FlowMesh $\ell_{1}$ , Reg. ( $\lambda=10^{-3}$ )	Velocity	$\bm{0.0323\pm 0.0186}$	$\bm{0.0114\pm 0.0055}$	$\bm{92.5\pm 2.5}$
	Temperature	$0.1350\pm 0.1675$	$0.0536\pm 0.0657$	$83.7\pm 12.1$
	TKE	$\bm{0.0142\pm 0.0095}$	$\bm{0.0050\pm 0.0031}$	$\bm{95.1\pm 1.9}$

Table 4: PWR sub-channel distributional metrics over six fixed, matched training seeds (mean

\pm

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 $\downarrow$	JS $\downarrow$	Wasserstein $\downarrow$	PDF Coverage (%) $\uparrow$
Flow Match – No Reg ( $\ell_{1}$ , $\lambda{=}0$ )	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 ( $\ell_{1}$ , $\lambda^{\star}{=}10^{-3}$ )	Velocity	$0.0175$	$0.0075$	$0.0255$	$\mathbf{94.40}$
	Temperature	$0.0310$	$0.0134$	$2.6921$	$\mathbf{93.07}$
	TKE	$0.0047$	$0.0018$	$0.0006$	$\mathbf{96.98}$

Table 6: PWR sub-channel single-run comparison of

\ell_{1}

and

\ell_{2}

flow-matching supervision and the autoencoder reconstruction ceiling. For each loss, the unregularized baseline (

\lambda=0

) is compared with the selected regularized setting (

\lambda=10^{-3}

). The autoencoder reconstruction diagnostic passes

N=4,000

reference fields through

D_{\psi}\circ E_{\phi}

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 $\downarrow$	JS $\downarrow$	PDF Coverage (%) $\uparrow$
FlowMesh No Regularizer	TKE	$0.003\,82$	$0.001\,42$	$97.800$
	Pressure	$0.004\,89$	$0.001\,77$	$96.162$
	Temperature	$0.186\,62$	$0.066\,22$	$80.288$
	Z-Velocity	$0.000\,38$	$0.000\,14$	$99.196$
	Y-Velocity	$0.001\,89$	$0.000\,70$	$98.586$
	X-Velocity	$0.003\,39$	$0.001\,34$	$97.784$

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	$6645.9000$	$-13\,974.3800$	$25\,704.1000$
	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

\ell_{1}

penalty on the latent velocity magnitude. The regularization sweep in Fig. 6 shows that increasing

\lambda

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

\ell_{1}

velocity regularization with

\lambda=10^{-1}

, whereas vanilla flow matching uses the identical model with

\lambda=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	$94.7\pm 11.6$ (adaptive)	250 (DDIM)
Inference cost per sample	$\sim$ 1.4 TFLOP	$\sim$ 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\times 64\times 64

), which is identical across datasets; training times were measured on the PWR subchannel latents, with FlowMesh at

\lambda=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

\lambda{=}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

\lambda\in\{0,10^{-4},10^{-3},10^{-2},10^{-1},1\}

. Under

\ell_{1}

supervision, KL decreases from the unregularized value of approximately

0.032

to approximately

0.022

at

\lambda=10^{-4}

and remains comparatively low through moderate

\lambda

, before increasing sharply at

\lambda=1

. The

\ell_{2}

model deteriorates at smaller regularization weights. (b) Empirical Jacobian-norm proxy

\hat{J}=\max_{(\mathbf{z},t)\in\mathcal{S}}\|\nabla_{\mathbf{z}}\mathbf{u}_{\theta}(\mathbf{z},t)\|_{F}

estimated on the sampled evaluation set

\mathcal{S}

. The proxy decreases from approximately

65

at small

\lambda

to approximately

28

for

\ell_{1}

and

32

for

\ell_{2}

at

\lambda=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

\lambda

. Small-to-moderate

\ell_{1}

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.

Flow Mesh: A Velocity-Regularized Flow Matching Framework for Stable Distributional Modeling of Physical Fields on Unstructured Meshes

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