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

TL;DR

This paper introduces FOT-CFM, a novel functional flow matching framework in Hilbert space that leverages optimal transport for efficient, resolution-invariant turbulent flow generation, outperforming existing models.

Contribution

The paper presents a new generative model for turbulence in infinite-dimensional function space using optimal transport, enabling faster, resolution-invariant sampling of complex turbulent fields.

Findings

FOT-CFM outperforms state-of-the-art models in reproducing turbulent statistics.

The approach achieves faster sampling without simulation, in a resolution-invariant manner.

Demonstrated effectiveness on Navier-Stokes, Kolmogorov Flow, and Hasegawa-Wakatani systems.

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.