Flow Straight and Fast in Hilbert Space: Functional Rectified Flow

Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensional settings. However, the extension of rectified flow to infinite-dimensional spaces remains unexplored. In this work, we establish a rigorous functional formulation of rectified flow in an infinite-dimensional Hilbert space. Our approach builds upon the superposition principle for continuity equations in an infinite-dimensional space. We further show that this framework extends naturally to functional flow matching and functional probability flow ODEs, interpreting them as nonlinear generalizations of rectified flow. Notably, our extension to functional flow matching removes the restrictive measure-theoretic assumptions in the existing theory of Kerrigan et al. [37]. Furthermore, we demonstrate experimentally that our method achieves superior performance compared to existing functional generative models. Related work Generative models have significantly advanced in recent years, with methods such as Generative Adversarial Networks (GANs), diffusion models, and flow matching achieving state-of-the-art results. GANs, introduced in Goodfellow et al. [23] , leverage adversarial training to generate high-quality samples from complex data distributions. Diffusion models, based on stochastic differential equations, iteratively remove noise from corrupted data through a learned denoising process, demonstrating strong generative capabilities [31, 74] . Flow matching constructs a path of conditional Gaussian distributions to interpolate the data distribution and a reference distribution [50, 8] . Rectified flows, introduced by Liu et al. [56] , offer a deterministic alternative to stochastic generative models by constructing straight transport paths between source and target distributions. In contrast to diffusion models, which rely on stochastic sampling, rectified flows enable more efficient and interpretable generation with reduced computational overhead. The associated straightening effect has been shown to facilitate high-quality generation with very few sampling steps [45] . Further theoretical development, such as its connection to Optimal Transport, has been explored in Liu [54] . Recent advances have extended rectified flow methods to a broad range of generative tasks, including image generation and editing [83, 69, 59, 79, 14] , 3D content creation [21] , text-to-speech synthesis and editing [25, 52, 26, 80] , audio reconstruction [81, 53] , video generation [78], and multi-modal generative modeling [46, 51] . Despite this growing popularity, existing rectified flow models are still constrained to finite-dimensional spaces. In this work, we address this limitation by extending rectified flow to general Hilbert spaces, thereby enabling modeling in infinite-dimensional function spaces. A key motivation for studying functional generative models is that many data sources are inherently functional-such as snapshots of time series or solutions to partial differential equations. A prominent example is neural stochastic differential equations (Neural SDEs), where neural networks are trained to model path-valued random variables that solve SDEs [38] . However, these models typically assume that the data follows an underlying SDE structure, as in financial time series [82, 33] , which limits their general applicability. On the other hand, representing finite-dimensional data as continuous functions offers several advantages: it naturally supports variable resolution, accommodates diverse data modalities using simple architectures, and enhances memory efficiency [16] . Motivated by these benefits, recent works have extended generative modeling beyond finite-dimensional Euclidean spaces to functional settings. Dutordoir et al. [17] and Zhuang et al. [85] adapt existing diffusion models to functional data by conditioning on discretized pointwise evaluations. A more general direction involves defining stochastic processes directly in infinite-dimensional Hilbert spaces, leading to the development of functional diffusion models [36, 20, 5, 60, 48, 49, 65, 27, 62] . These works extend score-based methods by studying diffusion processes over function spaces. Similarly, functional flow matching [37] generalizes the flow matching framework of Lipman et al. [50] to infinite-dimensional settings. However, as noted in their work, the analysis of [37] relies on strong measure-theoretic assumptions that are often difficult to verify in practice. In contrast, we extend rectified flow to Hilbert spaces under more tractable and verifiable conditions, providing a rigorous and broadly applicable foundation for functional generative modeling. Rectified flows on Hilbert space In this section, we extend rectified flows to infinite-dimensional Hilbert spaces, demonstrating the fundamental property of marginal distribution preservation. Let H be a separable Hilbert space. Given a probability triplet (Ω, S