Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Space

Li Kunpeng; Wan Chenguang; Qu Zhisong; Lim Kyungtak; Virginie Grandgirard; Xavier Garbet; Yu Hua; Ong Yew Soon

cs.LGApr 7, 2026

Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Space

Authors: Li Kunpeng, Wan Chenguang, Qu Zhisong, Lim Kyungtak, Virginie Grandgirard, Xavier Garbet, Yu Hua, Ong Yew Soon

Abstract

High-fidelity modeling of turbulent flows requires capturing complex spatiotemporal dynamics and multi-scale intermittency, posing a fundamental challenge for traditional knowledge-based systems. While deep generative models, such as diffusion models and Flow Matching, have shown promising performance, they are fundamentally constrained by their discrete, pixel-based nature. This limitation restricts their applicability in turbulence computing, where data inherently exists in a functional form. To address this gap, we propose Functional Optimal Transport Conditional Flow Matching (FOT-CFM), a generative framework defined directly in infinite-dimensional function space. Unlike conventional approaches defined on fixed grids, FOT-CFM treats physical fields as elements of an infinite-dimensional Hilbert space, and learns resolution-invariant generative dynamics directly at the level of probability measures. By integrating Optimal Transport (OT) theory, we construct deterministic, straight-line probability paths between noise and data measures in Hilbert space. This formulation enables simulation-free training and significantly accelerates the sampling process. We rigorously evaluate the proposed system on a diverse suite of chaotic dynamical systems, including the Navier-Stokes equations, Kolmogorov Flow, and Hasegawa-Wakatani equations, all of which exhibit rich multi-scale turbulent structures. Experimental results demonstrate that FOT-CFM achieves superior fidelity in reproducing high-order turbulent statistics and energy spectra compared to state-of-the-art baselines.

Figures & tables

Figure 1: FOT-CFM in infinite-dimensional function space, OT-aligned operator training and ODE Sampling

Metrics			DDPM	FFM	DDO	GANO	FOT-CFM
NFE=5	KDE	$R^{2}$	0.9897	0.9975	0.8833	0.8799	0.9982
	KDE	RMSE	0.0027	0.0013	0.0090	0.0092	0.0011
	RS	$R^{2}$	0.3941	0.9946	0.5552	0.8008	0.9953
	RS	RMSE	1.0088	0.0949	0.8643	0.5784	0.0892
	DS( $kx$ )	$R^{2}$	0.0508	0.9913	0.2712	0.7023	0.9919
	DS( $kx$ )	RMSE	1.0448	0.1000	0.9155	0.5851	0.0967

Table 1: Comparison on Kolmogorov Flow. We evaluate physical fidelity using spectral metrics (radial and directional) and density consistency metrics, alongside computational efficiency (NFE). Bold indicates the best performance.

Figure 2: Comparison of generative models on the 2D Kolmogorov Flow. Each row presents the results of a specific model (FOT-CFM, FFM, DDO, DDPM, and GANO) at a fixed inference budget (NFE=5).

Metrics			DDPM	FFM	DDO	GANO	FOT-CFM
NFE=5	KDE	$R^{2}$	0.8848	0.9892	0.9412	0.9593	0.9949
	KDE	RMSE	0.0283	0.0087	0.0201	0.0168	0.0059
	RS	$R^{2}$	0.1637	0.9536	0.4474	0.9149	0.9767
	RS	RMSE	2.0988	0.2817	1.2896	0.5062	0.2649
	DS( $kx$ )	$R^{2}$	0.0464	0.8326	0.1047	0.6609	0.8910
	DS( $kx$ )	RMSE	2.4661	0.5906	1.6312	1.0039	0.5692

Table 2: Comparison on Navier-Stokes Equations. We evaluate physical fidelity using spectral metrics (radial and directional) and density consistency metrics, alongside computational efficiency (NFE). Bold indicates the best performance.

Figure 3: Comparison of generative models on the 2D Navier-Stokes equations. Each row presents the results of a specific model (FOT-CFM, FFM, DDO, DDPM, and GANO) at a fixed inference budget (NFE=5).

Metrics			DDPM	FFM	DDO	GANO	FOT-CFM
NFE=100	KDE	$R^{2}$	0.2400	0.9856	0.9911	0.3412	0.9932
	KDE	RMSE	0.0629	0.0041	0.0046	0.0392	0.0038
	RS	$R^{2}$	0.8735	0.9878	0.9811	0.5673	0.9912
	RS	RMSE	0.5404	0.1377	0.1528	0.9995	0.1309
	DS( $kx$ )	$R^{2}$	0.7947	0.9704	0.9713	0.2922	0.9814
	DS( $kx$ )	RMSE	0.5851	0.1708	0.1517	1.0864	0.1121

Table 3: Comparison on the density

n

of Hasegawa-Wakatani Equations. We evaluate physical fidelity using spectral metrics (radial and directional) and density consistency metrics, alongside computational efficiency (NFE). Bold indicates the best performance.

Metrics			DDPM	FFM	DDO	GANO	FOT-CFM
NFE=100	KDE	$R^{2}$	0.1737	0.9905	0.9893	0.8976	0.9991
	KDE	RMSE	0.0709	0.0039	0.0054	0.0140	0.0016
	RS	$R^{2}$	0.5937	0.9021	0.9220	0.3023	0.9928
	RS	RMSE	1.3784	0.6765	0.6038	1.8062	0.1841
	DS( $kx$ )	$R^{2}$	0.3781	0.8266	0.8620	0.2007	0.9511
	DS( $kx$ )	RMSE	1.5505	0.8186	0.7303	2.1544	0.3931

Table 4: Comparison on the potential

\phi

Figure 4: Comparison of generative models on the density $n$ of 2D Hasegawa-Wakatani equations. Each row presents the results of a specific model (FOT-CFM, FFM, DDO, DDPM, and GANO) at a fixed inference budget (NFE=100).

Figure 5: Comparison of generative models on the potential $\phi$ of 2D Hasegawa-Wakatani equations. Each row presents the results of a specific model (FOT-CFM, FFM, DDO, DDPM, and GANO) at a fixed inference budget (NFE=100).

Flow-Based Generative Modeling

Flow-based generative modeling learns probability transports from simple noise to data for fast sampling, density estimation, and controllable generation.

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Spectrally Regularized Latent Flow Matching for Turbulence Generation

Khalid Rafiq, Aditya G. Nair

Department of Mechanical Engineering University of Nevada, Reno, NV, USA

Spectral Regularization Flow Matching Flow-Based Generative Modeling Computational Fluid Dynamics

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Latent diffusion and flow matching have emerged as leading approaches for synthetic turbulence generation, yet they systematically under-represent dissipation-range amplitudes. We introduce a latent flow matching framework with a spectrally regularized compression stage that directly targets this failure mode. On a 256^2 DNS dataset at Re_f \approx 2250, replacing an MSE-trained VAE with a zone-weighted log-spectral objective raises deep-dissipation retained spectral power from 25% to 94% in reconstruction and from 20% to 79% in unconditional generation. The improved latent representation also yields a substantially better sampling cost-fidelity tradeoff: the MSE-trained latent space imposes a fundamental quality ceiling near DD bias -0.70 that no integrator or step-count can overcome, while the spectrally regularized latent space reaches DD bias -0.117 at just 20 function evaluations. Mechanistically, encoder-decoder swap experiments show that the improvement is driven primarily by encoder-induced latent reorganization rather than decoder capacity, while a support-amplitude decomposition reveals that MSE-trained models behave as conservative suppression models, minimizing pointwise error by attenuating intermittent high-wavenumber structure. Both pipelines recover the second-order structure function and the correct sign of S_3, indicating the correct cascade direction without explicit supervision. A small residual gap in the magnitude of S_3 suggests that phase-coherent triadic organization remains a complementary axis to amplitude fidelity for future generative turbulence models.

Abstract

Flow-Based Generative Modeling

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