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

2026-04-07

AI Summary

Traditional deep generative models are constrained by discrete pixel representations, making it difficult to capture the continuous functional nature and multiscale intermittency of turbulent fields. This work proposes the FOT-CFM framework, which extends optimal transport–guided flow matching to infinite-dimensional Hilbert spaces for the first time, enabling direct modeling of probability measures over physical fields without grid discretization. By constructing a straight-line probability path between noise and data distributions, the method achieves resolution-independent, simulation-free efficient sampling. Experiments on Navier–Stokes turbulence, Kolmogorov flows, and the Hasegawa–Wakatani system demonstrate that FOT-CFM significantly outperforms existing approaches, accurately reproducing higher-order statistics and energy spectrum structures.

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.

Problem

• turbulent flow generation
• functional data
• Hilbert space
• resolution invariance
• multi-scale intermittency

Innovation

• Functional Flow Matching
• Optimal Transport
• Hilbert Space
• Turbulent Field Generation
• Resolution-Invariant Generative Modeling