Flow Straight and Fast in Hilbert Space: Functional Rectified Flow
Authors: Jianxin Zhang
Clayton Scott
Venue: ICLR 2026 Poster (2026)
Keywords: Hilbert space
superposition principle
Primary area: generative models
TL;DR: We generalize rectified flow to a separable Hilbert space and unify it with other functional generative model under a common framework.

Abstract: 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 \citet{kerrigan2024functional}. Furthermore, we demonstrate experimentally that our method achieves superior performance compared to existing functional generative models.

Published as a conference paper at ICLR 2026
FLOW STRAIGHT AND FAST IN HILBERT SPACE: FUNCTIONAL RECTIFIED FLOW
Jianxin Zhang
Clayton Scott
Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109 I Jianxin nz umi ch edu
Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109 el clayscot@uml wh..c.hh.oo edu
ABSTRACT
Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensionall settings, However, the extension of rectified flow to -inntt--iimesionall spaces remains unexplored. In this work, we establish rigorous functional formulation of rectified flow in an infiniteddimensional Hilbert space. Our approach builds upon the superposition principle for continuity equations in an infiniteddimensional space. We further show that this framework extends naturally to functional flow matching and functional probability flow ODEs, interpreting them as nonlincar generalizations of rectified flow. Notably, extension to functional flow matching removes the restrictive measure-theoretic assumptions in the existing theory of Kerrigan et al.] (2024). Furthermore, we demonstrate experimentally that our method achieves superior performance compared to existing functional generative models.
INTRODUCTION
Generative modeling has witnessed significant advancements, with methods such as flow matching
(Lipman et al., 2023), diffusion models (Song and Ermon 2020 Ho et al., 2020 [Song et al. 2021), and rectified flows (Liu et al., (022) achieving state-of-the-arr performance across various data types,
including audio Kong et al. (2021): (021::00 Goel et al. (2022), image Dhariwal and Nichol (202I) Kang et al.
(2023). and video Ho et al. (2022 Saharia et al. (2022).
Many of these techniques, originally developed inite-dimensional Euclidean spaces, have been extended to infinite-dimensional settings, as seen in functional flow matching (Kerrigan et al. (2024). functional GAN (Rahman et al., 2022) and functional diffusion models (Franzese et al. (0233] Sun et al. 2023 Bond- Taylor and Willeocks 2024 Na al. 2025). Functional models enable variable- resolution generation and simple, modality-agnostic architectures, making them effective across images, 3D objects, and scientific data (Dupont et al.) 2022. Kerrigan al., (202)). They can also improve training speed, sampling efficiency, and memory use (Bond- and Willcocks/ 2024 Franzese et al., 2023). However, despite the success of rectified flows in finite dimensions, their functional generalization remains an open problem.
In this work, we develop a mathematically rigorous framework for rectified flows in general separable Hilbert spaces. We further show that this framework naturally extends to proposed models of functional flow matching Kerrigan al. (2024) and function probability flow ODEs Na et al. (2025)
in Hilbert spaces, interpreting them nonlinear generalizations of rectified flow. Notably, our approach removes the restrictive measure-theeoetic assumptions required in Kerrigan al. (2024).
This theoretical foundation offers unified lens for functional generative models. Additionally, empirically validate functional rectified flow models and demonstrate their superior performance compared to other functional generative models.
Our contributions can be listed as follows:
We develop a theoretical framework that extends rectified flow to a general separable Hilbert spaces.

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We provide the generative insight that several existing ODE-based functional generative models, including Functional Flow Matching (FFM) (Kerrigan et al. (024). functional PF-ODEs (Na et al., 2025], and our Functional Rectified Flow, naturally arise instances of a single unified formulation.
We provide empirical validation across images and PDE dynamics, demonstrating that FRF achieves competitive superior performance compared with prior functional models under matched architectures.
The remainder of the paper is organized as follows. Section 2 reviews related work on rectified flows and functional generative models. In Section 3 we present main theoretical result, which lifts rectified flows to general Hilbert spaces. Section discusses connections to functional flow matching and functional probability flow ODE. Architectural choices for functional rectified flow models are discussed in Section Experimental results across several domains are reported in Section Finally, Section 7 concludes with a summary of findings and discussion of limitations and future directions.
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 results. GANs, introduced in Goodfellow et al. (2014), leverage adversarial training generate high- quality samples from complex data distributions. Diffusion models, based on stochastic differential equations, iteratively transform from noise to data via a score function approximated by minimizing quadratic objective (Vincent, 2011). demonstrating strong generative capabilities (Ho et al.) 2020 Song and Ermon, 2020; Song et al. 2021). Flow matching constructs a path of conditional Gaussian distributions to interpolate the data distribution and a reference distribution (Lipman et al. 2023
Chen et al./ 2018). These approaches have achieved state-of- oftte-of-thh-art performance across diverse modalities, including audio (Kong et al., [202]] Goel et al.) (022). images (Dhariwal and Nichol 2021 Kang et al. (0 and video (Ho et 2022 Saharia et al. 2022). Recent advances (Albergo and Vanden-Eijnden 2023) Rout et al. 2024) further show that deterministic flows and SDE-based diffusion models can be understood within a unified framework.
Rectified flows, introduced by Liu al. 2022), offer a deterministic alternative to stochastic gen- erative models by constructing straight transport paths between source and target distributions, In contrast 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 (Lee et al., 2024). Further theoretical development, such as its connection to Optimal Transport, has been explored in Liu (2022). Recent advances have extended rectified flow methods to a broad range of generative tasks, including image generation and editing (Zhu et al. 2024 Rout al.) 2024 Ma et al., 2024 Yang et al. 2024) Dalva et al.
2024), 3D content creation (Go et al. 2024).
texl-to-speech
synthesis and editing (Guan al. 2024 Liu et al. 2024b Guo al. 2024 Yin et al.
2025). ) audio re-
construction Yuan et 2025 Liu et al, 2024c). video generation [Wu et al. 2024). and modti-modal
generative modeling (Li et al. 2024 Liu al., 2024a).. Despite this growing popularity, existing rectified flow models are still constrained to finie-dimensional spaces. In this work, we address this limitation by extending rectified flow to general Hilbert spaces, thereby enabling modeling in infiniteddimensional function spaces.
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 (Kidger et al.) 202Tp. However, these models typically assume that the data follows an underlying SDE structure, as in financial time series (Zhang et al.) 2025) Issa et al., 2023), 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 (Dupont et al. 2022). Motivated by these benefits, recent works have extended generative modeling beyond dimee-iimensionll Euclidean spaces to functional settings. Dutordoir et al. (2023) and Zhuang et al. (2023) adapt existing diffusion models to functional data by conditioning on discretized pointwise evaluations.
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A more general direction involves defining stochastic processes directly in
Hilbert spaces, leading to the development of functional diffusion models (Kerrigan et al. 2003
Franzese et al. 2023] Bond- and Willcocks) 20244 Mittal et al. |022] Lim et al.) [223a b
Pidstrigach et al. 2024 Hagemann et al. 2023] Na et al.
2025). These works extend score-hased
methods by studying diffusion processes over function spaces. Similarly, functional flow matching
(Krrrian et al. 2024) generalizes the flow matching framework of [ipman et al. (2023) to infinite-
dimensional settings. However, as noted in their work, the analysis of Kerriga et al. (2024) relies
on strong -eeasuee-hheoeete assumptions that are often difficult to verify in practice. In contrast, we extend rectified flow Hilbert spaces under more tractable and verifiable conditions, providing a rigorous and broadly applicable foundation for functional generative modeling.
3 RECTIFIED FLOWS ON HILBERT SPACE
In this section, we extend rectified flows to infiiiteddimensional Hilbert spaces, demonstrating the fundamental property of marginal distribution preservation. Let H be separable Hilbert space. Given a probability triplct (92 S P). denoting the sample space, sigma algebra, and probability measure, we take the perspective that the law of stochastic process X [0. 1] x Ω + H is the distribution of the function-
vanlue valued random variable defined as w + X(: w.). For simplicity, we denote the value of X at time E [0, 1] by X2 and the associated marginal measure by pe-- A random process X will thus also be denoted { {X,, )e or simply (Xe}e. Unless otherwise specified, (.,-) refers to the inner product in Ht and the norm ||-|| refers to its induced norm.
3.1 DEFINITIONS OF RECTIFIABLE AND RECTIFIED FLOWS
Given process X. the induced rectified flow is a random process (Z) that is defined in terms a quantity called the expected velocity of X. Definition 1. Let X =
be a pathwise continuously differentiable random process on separable Hilbert space H. The expected velocity of X is defined
xx (1, x) = I2 Xx X, =]. Vo E supp(XX.),t E [0
and is set to zero if x & supp(X.). Here, X, denotes the time derivative of Xt, and supp(X,) the support of the random variable X₁-
Let C' denote the space of continuously differentiable functions. Pathwise continuous differentiability not data assumption but a benign restriction on the design of X. i.e., 1 + X, needs to be C' with probability 1. For an interpolation process X, Xi + Xp. the pathwise continuous differentiability condition is satisfied by taking both 0e and 3, to be real-valued C'l functions.
An initial value problem (IVP) is differential equation with specified initial condition for the unknown function. To ensure the well-posedness of rectified flows in Hilbert space, we assume that the corresponding IVP admits unique solution and a continuous solution map.
Assumption 2. Given a function v [0, 1] x H + H. the initial value problem
z(t)= = + J/' ((s,2(s))dss u E H.
(I)
admits a unique solution 2(-) E C² ((0. 1]; H). Furthermore, the solution map tb H. C H) mapping from the initial value to the solution path, defined by 4uu) is a continuous mapping from (H. (x) to (C" 1]; H), ll
In Proposition 18 in Appendix, we show that Assumption 2 holds under some standard Lipschitz condition. We emphasize that Proposition 18 is presented as a convenient sufficient condition rather 2 can in principle hold under other regularity conditions.
than as a core assumption, since Assumption In the following assumption. we enforee that xx x) has finite integral in time to ensure that the process does not exhibit pathological behavior, such as infinite total drift.
Assumption 3. Given a pathwise continuously differentiable random process {Xe
1.' J 11 ((x))))))) < co, and E
Xx|
sup 1E(0.1]
< <0,
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where h4 denotes the distribution of X₁ E for each E [0, 1].
With these notions, we now define rectified flow.
Definition 4. We say that a stochastic process (et)a rectifiable if Assumption holds for (1, x)) and Assumption 3 holds for X. In this case, the process defined by
Z, Zo S.¹ (s, Zo Xo:
called the rectified flow induced by X
Unlike diffusion models, which rely on stochastic differential equations (SDEs), rectified flows are defined in terms of deterministic ordinary differential equation (ODE) guided by o* (t,
3.2 THE MARGINAL PRESERVING PROPERTY
A key property of rectified flow is that it preserves the marginal distributions of the original stochastic process. This is formalized the following theorem which generalizes Liu et al. (2022) from finte-ddmensional Euclidean space to separable Hilbert space H.
Theorem 5. Assume that an H-valued stochastic process {X+} is rectifiable, and let {Ze} be the induced rectified flow. Then, for all t [0, 1]. Z, _ Xt-
The central challenge in proving the theorem lies in developing the superposition principle within a general separable Hilbert space setting. Informally, the superposition principle says that any distribution solving the "continuity equation" driven by xx can be "disintegrated" into distribution over paths that solve the associated ODE (2). The proof requires establishing several nontrivial technical results and rigorously connecting the resulting theorei- decomposition to the solution of Equation (I). We defer the exact statement of the superposition principle and the proof to Section in the appendix. In Appendix Section E we empirically verify Theorem on synthetic data where Xg and X2 are sampled from tractable distributions and Ex can be computed in closed form.
Note that Ze derives its randomness entirely from the random initial condition Zo. Intuitively, this result states that while Z, evolves deterministically given Zo according to an ODE, it preserves the marginal distributions Xa corresponds to a simple reference distribution (e.g., i.i.d. Gaussian noise) and X, represents the target data distribution, then learning the velocity field ex enables sampling from the data distribution by solving ? with randomly initialized Zo.
X,. When X, is constructed interpolate between two distributions, where
The rectified flow method of Liu al. 2022) corresponds to specific choice of X (X.) given by
X, LX + - 1)
where Xg and X2 are independently sampled from the noise and data distributions, respectively. This formulation was shown to be effective in miite-dimensional Euclidean spaces.
If we define X, = tX1 + (1 - X) as in (3. where Xo. X1 E H, then X, = X1 - Xo. The velocity field xx (t, modeled by neural network up. can be trained to minimize
mjn L [aaxx((< 2o) - vo( ())
where x is a realization of the stochastic process and is the value of x at time t. Note that the objective can be approximated by randomly sampled data X1, noise Xo, and timestamp L.
For sampling, we numerically integrate the ODE: Z, = Zo + fh (s(s, Z, ds, Zo ~ Xo, allowing us generate samples from X2 by evolving from Xu using the learned expected velocity vg- This approach computationally more efficient than stochastic sampling methods used in diffusion models, as it avoids the nuances of reversing an SDE.
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3.3 PROPERTIES OF RECTIFIED FLOW
The desirable properties of rectified flows in finiteddimensional settings, as discussed in Liu et al. 2202)-includingg transport cost reduction and the straightening effect-also hold in the infinite- dimensional Hilbert space setting, These results follow directly from the maaginnal-preservinn property established in Theorem S using arguments that largely mirror those in finite dimensions. As the extensions mainly involve technical but routine bookkeeping, we defer the detailed statements and proofs to Section B in the appendix.
4 CONNECTIONS TO OTHER FUNCTIONAL MODELS
We place our work in the broader context of functional generative modeling by examining its relationship functional flow matching (Kerrigan al. 2024) and functional probability flow ODE (Na et al. 2025). Our framework accommodates interpolation paths of the form:
X = a + 3,Xo:
(5)
where Ote, 3r are continuously differentiable functions of time t. This generalized parameterization recovers rectified flow when a₁ 33 the function rectified flow. We show that both functional flow matching (Kerrigan et al. 2024) and
We call Equation (5) the nonlinear extension of
functional probability flow ODE (Na et al. (025) can be viewed as nonlinear extensions functional rectified flow within our general framework.
To facilitate our discussion, we first define the Gameron-Martin space and the notion of Wiener process in Hilbert space. Let Q be trace-class, positive, symmetric operator on H. The associated Cameron-Martin space is given by Ho Q² (H), the range of Q², equipped with the inner product
(I,4)
A Q-Wiener process W, is continuous Gaussian process in H satisfying Wo = 0 and having independent, stationary increments, with W, W, N'00 (1 s)Q)
U)H Hi == {0 (Q- txx Q 7
for all 0 S 8 S
4.1 COMPARISON WITH FUNCTIONAL FLOW MATCHING
By examining Equation (6) in Kerrigan et al. 2024), we can directly see that Equation (5) subsumes their proposed flow matching models as special cases. In Table [ we summarize how the "VP" and "OT" paths introduced in their work arise as specific instances of (av,, 8.).
The framework of Kerrigan et al. (2024) relies on strong measure-theoretic assumption that may not hold even in dinen-iameniionall Euclidean space. Suppose there exists stochastic process (X.)tejo.1] interpolating Xo (noise) and X₁ (data), with μ, denoting the marginal law of Xt, and !r, the conditional law of X, given X; æ. |Krrrgann et al. (2024) assume: fi < 14, i.e., the conditional measure fu must be absolutely continuous with respect to the marginal μls for almost every As admitted by the authors in Kerrigan al. (2024), this assumption difficult satisfy and verify, and generally fails even in finite-dimensional Euclidean space. If X, is a trace-class Gaussian process, this reduces to the requirement that X, lies in the Martron space of the
initial Gaussian measure, i.e., X1 E c. c//2(()) (7() when X, ~ (0, Co). As also noted in Kerrigan et al. (2024), even this condition is highly restrictive and challenging to verify in practice. In section D in the Appendix, we show that this assumption would fail in the trivial case where X3 and Xn are gaussian and equal in distribution. Although assume that the processes differentiable, this assumption reflects a design choice rather than a data assumption.
pathwise continuously
Table 1: Functional flow matching paths proposed in Kerrigan et al. 2024 are special cases of
Flow Path
3
OT (Optimal Transport)
t
1- (1 o )., aiin (0,
VP (Variance Preserving)
arbitrary E [0, / o
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4.2 COMPARISON WITH FUNCTIONAL PROBABILITY FLOW ODE
Throughout this section, we adopt the convention that time = 0 corresponds to the data distribution and to the noise distribution. This the reverse of the convention used in the rest of the paper, and is chosen to align with the standard formulation of score-based generative modeling and SDEs. This section also references technical concepts (c.g., Q-Wiener process, Cameron- Martin space) that will be defined in the appendix.
The concurrent work of Na et al. (2025) introduces an analogue of the probability flow ODE from Song al.] (202I), adapted to infinite-dime diinite-dimensionall Hilbert spaces. Specifically, they consider the following variance- variance-preservee preserving SDE in H:
at Yd for JW, Yo ~ Paita:
dY, = -
where W, is a Q- Q-Wiener process in H. a a bounded, continuous function taking values on Rz0- Let {G: denote an orthonormal basis of H consisting of eigenvectors of Q, and let Ho denote the corresponding Coameron-Martinn space. To support the analysis, we recall the formal definition the logarithmic gradient.
Definition 6. A Borel probability measure μ is said to be Fomin differentiable along h E Hq there exists a function Ph (L, such that for all cylinder functions of the form
F(x) ((C1,2),.... ()) E CØ (R m m N,
we have
[ on - F(x) (x) u(dx),
where On denotes the Gâteaux differential along h. Let K. be a subspace of H. If there exists function P%c H. H. such that
(d)()), M) = (x) for all H. h E,
then dhe called the logarithmic gradient of μ along K.
During training, a score network S(t, Y2) is learned to approximate the logarithmic gradient by minimizing the objective
L.
S(t, Y.) - Pa1o (Y:)
(Y)) dt.
where Mijx, denotes the conditional distribution of Y, given Y,. Let S denote its minimizer.
For sampling, Na et al. 2025) propose following probability flow ODE, to be solved in reverse
time from tot 0:
0t (Y, 2 O ((.)) di, Y ~
dY, =
(6)
The following proposition shows how this ODE fits within rectified flow framework. Proposition 7. Equation
(6) is the rectified flow induced by the process Y: defined
= vn( Yo Vn(()U,
o. ds). x(t) = S/ exp (-f" o, ds, and Y! ~ Pait U ~ \O Q) sampled independently. Equivalently, B|y; Y (Y) S (t, Y!))
where rtt) = exp (-
Thus, the probability flow ODE (6) corresponds the time-reversal of the nonlinear rectified flow in
Equation (5). with the interpolation weights given by o, 17(1 and 3, Vs(1 t).

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5 ARCHITECTURES FOR FUNCTIONAL GENERATIVE MODELING
Let H. be Hilbert space of functions from some coordinate space M to F. To implement functional rectified flows in practice, one must approximate the expected velocity field e² (xe, H [0, 7] + H. which is inherently defined over an
domain. Directly learning such a mapping
generally intractable. However, if H Ly under the conditions of Theorem 2 in Franzese et al. (2023), any x E HH can be fully characterized by its pointwise evaluations {< J. where p: M. This insight enables the design of networks that act on sampled representations rather than abstract functional inputs, thus making implementation feasible. Following Franzese ctt al. (2023) and Kerrigan ct al. (2024). we consider three practical architectures for this purpose: Implicit Neural Representations (INRs), Transformers, and Neural Operators. We include this discussion of architectural choices for completeness but refer the reader to Franzese et al. (2023) for
comprehensive treatment.
The first approach builds on implicit neural representations (INRs) Sitzmann et al. (2020) and the model-agnostic meta-learning (MAML) paradigm Finn et al. (0177). Following the modulation strategy introduced for functional diffusion processes by Franzese et al. (2023), we represent the velocity field uo (z,, t) with a network n( t, 0) M + Et where is the shared base parameter and ab is a sample-specific modulation vector that is adapted online to encode each Iy. Formally,
(o t) = nlv" ,t,",tt,0 0), 3= = arg min
Σ (n t, 0pp)p xx(p:l)?? P.
(7)
For every Tr we initialize 1/- O and perform a small number of gradient-descent steps, with 0 fixed, to minimize the residual 77. During training, the global parameters 0 are updated via the loss 44) whereas v is re-optimized from scratch for each zt, enabling the network to have a functional representation while operating on finite discretizations {z, (p.].....
Our second approach employs transformer architectures (Vaswani et al. 2017), treating discretized function evaluations as sequences with positional information. For each Tt. we consider the set of finite discretizations [x.[p:]]} as input features and their corresponding coordinates {p.} positions. Each 2t [px] is embedded into a higher-dimensional vector space, and summed with the positional encodings of Pi, resulting in sequence of embeddings {w: }. This sequence is then processed by We note that Cao (021)) 1 interpret transformers as mappings
[. }- between Hilbert spaces, and and Kovachki et al. (2021) discuss the learned rather than guaranteed nature of resolution invariance of transformers. While these theoretical perspectives are relevant, we do not explore them further and refer interested readers to the original papers for a detailed discussion.
transformer to obtain
Neural operators (Kovachki et al. | Kossaifi et al., (2024) offer principled framework for mod- cling functions in feihitt-dimensionaa settings and have demonstrated success in various applications (Peng et al. 2024; Qin et al. 2025). A neural operator learns a mapping G : H + H. trained using finite evaluations of functions. Once trained, it maps a uniform discretization ((x(px). P.)) to output values at the same locations, producing (G(x) [p.]]. In our implementation, we directly parameterize the velocity field Up using a neural operator. Similar to the transformer- basso approach, the positions {px} are encoded via positional embeddings and appended to the embeddings of the corresponding [p]]. Although neural operators typically require inputs to be defined on fixed, uniformly spaced grids and produce outputs the same grid locations- lemmttigs-limmiing flexibility for irregular adaptive sampling-they have been shown to be highly effective for PDE data (Kerrigan et al. 2024). [Bond]
Taylor and Willcocks (2024) implement an infinite-dimensionall generative model using neural operator-basee based architecture but rely on specialized diffusion autoencoder (Preechakul et al. 2022
making their approach highly specialized and not applicable to our setting.
EXPERIMENTS
We evaluate the proposed Functional Rectified Flow (FRF) model on three cense) (LeCun et al. 2010), CelebA 64 x 64 (CC BY-SA 4.0) (Liu et al. 20155). and the Navier-Stokes dataset (MIT License) (Li al. 2022), using INR, transformer, and neural operator implementations, Hilbert functional gen-
(MIT Li-
respectively. Our aim is to assess whether extending rectified flow to spaces yields competitive or superior performance compared with erative models across different architectural classes. Importantly, in Section 6.1.1 (INR), Section
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6.1.2 (ViT), and Section 6.2 (Neural Operator), as FDP and FFM, adopting the architectures from their original papers to avoid bias. Thus, the performance gains observed in experiments arise solely from the proposed functional flow matching objective, not from increased model capacity. For clarity, we present results two parts: (1) image-generation experiments mapping coordinates in [0, 1J2 to pixel values in IR and IR' respectively, and (2) PDE experiments on the Navier-Stokes dataset, Additional experimental details appear Section C of the appendix.
exactly the same network architectures
MNIST and CelebA, where each image is considered
function
6.1 IMAGE DATASETS
For image datasets, we focus our comparisons on Functional Diffusion Processes (FDP) (Franzese al. 2023) and Dc-DIFF (Bond-Taylon and Willcocks 2024), which are among the most competitive methods for image generation in functional settings, To ensure a fair comparison, we adopt the exact same architectural design as Franzese al. (202)), without introducing any specialized enhancements. FRF is implemented in JAX (Bradbury et 2018), building directly on the released codebase of Franzese al. (2023) (Apache-2.0) License). For INR-based models, modulation codes are optimized using steps of gradient descent per sample. For both MNIST and CelebA experiments, Xo is sampled from the stationary distribution of the functional SDE (9) in Franzese et (2023).
To evaluate generative performance, we consider the Fréchet Inception Distance (FID) Heusel et al., 2017) and the FID-CLIP score (Kynkäänniemi et al. 2022). where lower values indicate higher visual fidelity and closer alignment with the target data distribution.
6.1.1 INR ON MNIST
On MNIST (32 x 32), benchmark INR-based FRF model against the Functional Diffusion Pro- cess (FDP) (Franzese et al. 2023). We specifically adopt the INR architecture to test the effectiveness of FRF on lightweight models, as INRs naturally have a small parameter footprint while retaining strong functional approximation capabilities.
As shown in Table our INR-based FRF achieves a lower FID score than FDP while using the same number of parameters, indicating that functional rectified flow leads to improved sample quality even with lightweight models.
Figure D further illustrates the qualitative advantages of FRF. Notably, the super-resolved samples produced by our model at both 64 64 and 128 128 resolutions (Figures (b) and (d)) exhibit smoother digit contours compared to the naïvely upscaled real MNIST images (Figures (c) and (e)). This highlights the model's ability to generate coherent high-resolution samples by leveraging its continuous functional representation while being trained on lower resolution.
Table 2: Results on MNIST (32 x 32) using INR.
Method
FID (L)
Params
FRF (INR, ours)
0.41
0.1M A 0.1M
FDP (INR)
0.43
3
9
0
9
1
5
/
3
O
G
4
S
0
/
9
5
3
7
9
6
0
3
0
5
6
3 2
7
/ 5
7
8 9
0
6
8
9
0
4
3
4
5
9
7
1
3
4
2
6
/
4
/
8
5
1
9
2
3
(a)
(b)
(c)
(d)
(c)
Figure 1: Qualitative results MNIST: (a) samples generated at the original 32 32 resolution; (b)
super-resolved samples at 64 x 64; (c) real MNIST images upscaled to match (b); (d) super-resolved samples at 128 128; (e) real MNIST images upscaled to match (d).
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6.1.2 TRANSFORMER ON CELEBA
On CelebA (64 x 64), we evaluate transformer-base based FRF models and compare against FDP (both
INR and transformer variants) (Franzese al. 2023), FD2F (Dupont al. 2022), and oc-DIFF (Bond-Taylor and Willcocks/ 2024). We specifically adopt the transformer architecture to demonstrate that FRF capable of generating high-quality and visually compelling images on complex datasets.
Table Shows that FRF with a transformer backbone outperforms other functional generative models in terms of both FID and FID-CLIP scores, while also being significantly more parameter-efficien than oc-DIFF. Figure 2 further illustrates the visual quality of samples generated by FRE The model successfully captures rich facial details and produces sharp, high- sidelfidliity images.
Table 3: Results on CelebA (64 64) using Transoormr-bbaee architectures.
Method
FID (4)
FID-CLIP (1)
Params
FRF (Vision Transformer, ours)
6.63
3.70
2 20M
FDP (INR)
35.00
12.44
1M
FDP (Vision Transformer)
11.00
6,55
~ 20M
FD2F
40.40
10M
oc-DIFF
4.57
100M
Figure 2: Qualitative results of functional rectified flow with vision transformer.
6.2 NEURAL OPERATOR ON MAVIER-STOKER DATASET
We evaluate the proposed FRF model with a neural operator backbone on the Navier-Stokes dataset, benchmark consisting of solutions to the Navier-Stokes equations on 2D torus. This dataset captures complex fluid dynamics, making a challenging testbed for functional generative models. To be consistent with prior work (Kerrigan et al. 2024 condition Xo. Our code built upon the open sourced codebase of Kerrigan et al. (2024) (MIT license).
a -Mtttr kemell to sample the initial

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We compare our model against several baselines: the Denoising Diffusion Operator (DDO) (Lim et al. 2023a) using the NCSN noise scale, GANO (Rahman et al., (022). Functional DDPM (errigan
al. 2023), and Functional Flow Matching (FFM) (Kerrigan et al.] 2024).
Following Kerrigan et al. (2024), we evaluate the model performance by Density MSE, which is computed by applying pointwise Kernel Density Estimation (KDE) 1,000 generated and 1,000 real samples, followed by computing the Mean Squared Error between the resulting estimated densities. This metric evaluates how well the generated samples reproduce the overall spatial distribution of the data.
As shown in Table the FRF model achieves the lowest density MSE among all methods, indicating that the generated samples accurately match the distribution of real samples in the spatial domain.
Table 4: MSES between the density of the real and generated samples of the Navier-Stokes dataset.
Method
Density MSE (Mean + Std)
FRF
2.39 10
+ 4.45 X 10-6
FFM
4.50 x 10-5
+ 1.52 x 10-5
DDPM
1.02 10-4 + 8.20
10 10-6
DDO
9.61 x 10-3 + 1.26 x 10-2
GANO
4.16 x 10-3 - 1.82 x 10-3
7 CONCLUSION, LIMITATIONS, BROADER IMPACT, AND FUTURE DIRECTIONS
We have introduced functional extension of rectified flow by lifting it to general Hilbert spaces. Our theoretical results demonstrate that the marginal preserving property is preserved in the infinite- dimensional setting. This provides principled and tractable foundation for functional generative modeling. We demonstrate experimentally that our method achieves superior performance compared to existing functional generative models. Notably, while most competing approaches develop specialized architectures tailored to their methods, we demonstrate the flexibility of our framework by applying it across three distinct model architectures-eate originally designed for one of our competitors. Although our framework is broadly applicable, domain-specinn architectures and inductive biases may still be required for optimal performance high-complexity tasks.
As with all generative models, risks such as misuse in synthetic media generation remain, including the potential for creating misleading harmful content. We encourage responsible use and further research into interpretability. robustness, and safety in functional generative models to mitigate these concerns. At the same time, functional generative models offer significant positive societal impacts, including advancing scientific discovery through improved simulation of complex physical systems and supporting creative industries with new tools for high-quality content generation.
A natural theoretical direction is to extend the rectified flow framework to general Banach spaces. The key obstacle is the reliance on orthonormal bases in the current analysis, which are available in Hilbert spaces but absent in generic Banach spaces. On the empirical side, important extension is to develop functional versions generative models that operate on a broader class of data modalities, such irregularly sampled time series and 3D geometric objects.
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A PROOFS AND REMARKS
In this section, we present the proofs together with remarks on technical details.
A. REMARKS ON THE EXISTENCE OF THE VELOCITY FIRLD
Under our setting, the existence x follows from standard results.
Proposition 8 (Proposition 1. 1.10 of Da Prato and Zabezyk (1992). Assume that E is separable Banach space. Let be a Bochmer integrable E-valued random variable defined on (St,S, S,P) and let G be a o-field contained in S. There exists unique, up to set of IP P-robbability zero, integrable E-valued random variable R, measurable with respect to G such that
JA VV dP / RdP, VAEG.
The random variable is denoted as EV[ [G] and called the conditional expectation of V given G.
In our case, the state space H. is a separable Banach space. Hence, it suffices to assume X || < co for integrability. Under this condition, B[X, oXx)] exists and is o o (m))suuuuuaall.. By the Doob-Dynken lemma, there exists a measurable function t2 H. - H. such that
E|X, (Xx)) - (X.).
Defining v² x) = (x( for x E supp( X2) and (1, x) otherwise (which affects only a set of measure zero) yields the desired vector field. For completeness,
recall the Doob-Dynkin lemma: Lemma 9 (Lemma 1.14 of Kallenberg (2021)). Let f.g be measurable fanctions from (S, A) into measurable spuces () and (T,7), where S is Borel. The following are equivalent:
1. f is g-measurable, i.e. off) (g))).
2. there exists measurable mapping h T - S such that f h g-
This shows that the velocity field xxx is well-defined under mild integrability conditions.
A.2 DEFIIIIOOSS AND TECHNICAL LEMMA
In this section, we present definitions and technical lemmas for the invocation of the superposition principle in Hilbert space. We first review Fréchet differentiability
normed space.
Definition 10. Let and F be normed vector spaces, O an open subset of F containing 0. The derivative of E + at x E O a continuous linear map D&f from E to F such that
f(r + h) f(x) D&f(h)
lim b++
When E. H and F D&f f H" where H is the dual space of H. By the Riesz representation theorem, there is a unique E H such that D. (.,g), and we denote such by V (x(.). We define the second-order derivative V²f(2) the derivative of & E H + Vf(x) H.
Definition 11. Let 7 be separable Hilbert space with orthonormal basis {e:} and yp H + I. Define its asymptotic Lipschitz constant (x) at x H by
Lo(x) = lim Lo(x, E), c-0+
where
Ie(n)) - (p()) ||u - 2l
(o( sup (.vEB.())
Notice that Tup is upper semicontinuous Stepanov and Trevisan 2017).
Lemma 12. If :p: + R is a tifie-differentiaable function with bounded second-order derivative,
then Tnp (x) [[](()]]
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Proof. Fix and H. We first show (x(x) 1 [[Vp(=)]]
Let a, E (z) and b * a. By the mean value theorem (Theorem 3.2 of Coleman (2012)
(bb) ((()) (Vp(c), a) = (V b a) + (Vp(c) ( a)
where c to + (1 t for some E [0, 1]. Hence,
l(p(b) - x(a)] < V all + VQ(cc - V=(())llb-aa)
- V all +
sup
H.H 11b - allllc - x (12))1111)
We obtain the second inequality by applying the mean value incquality (Corollary 3.2 of Coleman (2012) to Vy. Suppose V2p is bounded by a constant Bg. Note that both c, c' E (x)).. Hence,
(p(b) s(a)]
- <|V+(x))] + Boc
|b all
The choice of a, is arbitrary, so L e) S [|Vp(())]| + Bie and Lp(x) S [Vp(()]..
Now it remains to show (x) 2 [/](()) WOLOG, we assume Vip(z) # 0 and let e [Vp(=)]]
Ve(a)
Let y/ = x + n UU Rn = I,, n N.
Fix a > 0. Then NN E N such that Vn 2 N. E Be(x)) and by definition of Fréchet differentiability
1 [) - ())) (RS(O), 1 n p0x + n
S
72
Hence, x0( u) ((x) 2 (Vx(x), u0 - - [[(()= For yn E (z).
Txp e) 2 1p(Un) (p(2n)
1// 2n0
|0(z + v) - (p(x))
1
( + 1 n. 1) - (0(x))
n
| a.
Since is arbitrary, (x( <) 2 [V)(()) Therefore, L (x) = [V)(())
Definition 13. Let A be the cylindrical functions on H. i.e..
4=fy E H y = o Pa, d is a non-negative integer, & E C (RR) has compact support)
where Pa(x) ((x, ez),..., (ar, ea)) is a finite dimensional projection and C (TR.) is the set of compactly supported function from IRd to R.
is an algebra (i.e., set closed under pointwise linear combinations and products), and a constant function trivially an element in A by letting d 0. By Lemma 12 L |Vp((>) II, Vi A. Let U, H + H. for t E [0. 1] be vector field. o2 defines an operator V : A + R acting on elements in A, i.e., for & E A. (V))) (Ve(z), (x))) Hence, V, satisfies V.(.f) = fVilg) + Vf...).
Also, [[v)))==)]= = (V) (x)) S (L))
We have to show one more lemma to invoke the results of (Stepanov and Trevisan 2017).
Lemma 14. Let be a Lipschitz, bounded function on H x [0, 11, with metric ((x, 1). (y, s)) +
(||=: yl + It 10-88 2/2)). Then there is a sequence of uniformly bounded functions (19k) on H x [0
that are also Lipschitz with uniform Lipschitz constants, such that (p((,tt E A for E [0.1] limg sp = ip pointwisely, and
lim sup (xpu(.))(z) S (xp(a))()) ((r,,) E H x [0.1].
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Proof. Let {ex} be orthonormal basis of H. Let Cip be the Lipschitz constant of yp.
(r, For integer & > O, recall that Ps: H + TR*, (a(x) = ((r, (2) ex)). Define (x))) H H. ne (2) Ei (2, eg)e; and tha R& [0. 1] R. (2 (EE 1e)) Note that (xr( - Pa(W))lR= = [lx(()) TH (v)ll S || - vll and VN (P)()). = (T))
Also note that
1/2
k
(x(r (F)(F S C+
Σ i=1
r;cs Σ Frc
+ (t - s)²
1=1
Fl + (1 - 8)²) 1/2
= Cp
||r
So thk is also CppLiipshitz.
Fix an integer l > 0. We first approximate the using standard mollification approach. Following Appendix C.5 Evans (2010), we define the standard mollifier function for e 0:
Ne :R + ne()) c "
where 7 R* + IR E C
Cexp 10
if || g+ < 1 if ||r| l+ 2
n(r)
(mp-1)
and the constant C selected so that Jun = 1. Let (r) the e-ball in R* around r and note that 7) is supported on B. (0) and Jg= The 1.
Define the mollification of v/e as lAs,e (r; 1) == Jax ne (y) - y, t)dy. Theorem 7 from Appendix C.5 of Evans 2010 asserts that he. C0000(1 Note that
1 ne(()(ex v,t) s/olF y, s)
hr,e(r,t) t) (F))
/ ne (c)lvul(r - v.t) ~x - v. s)
Rt
(v) [1r-Fl66 + (t =s) dy
Rt
Ci -Ir FllEe + (t s)3) &
and
/ (e v.t) -z) -vet)) t))y
(r, (r(r, t)
< Cul
B7(0)
< C.p.
Therefore, hg. is also Cop-Lipschitz and uniformly converges to t/k at a rate Ose when 6 - 0.
We can then extract sequence of smooth functions vh.. approximating 4 by letting en C,n and Llk.n = hk.c.- Note that e/k., E C (R+) is C--LLiscchit and || vhelloo < 1 n.
1
Let x R + R t (R) be smooth cutoff function such that x(s)
on [-1,1],
x(s) E [0, 1] on e (-4, -1) U (1, and X(s) = 0 on (-o0, -4] U [4, oo). Note that X and its derivatives of any order supported on -4.4).
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We now ready define the sequence ypu 7/ [0. + R by
(
(x(x;)) -
(Pa())) 122
Hence, (Px(s)) E A since u + x
Iwll
and VANA are smooth, u +
is compactly
supported, and fpu depends on z only through Pa(x). It remains to verify the following assertions:
1. iPk uniformly bounded.
2. 4Pk uniformly Lipschitz.
3. *Pk converges sp pointwisely.
4. limsup; (z) ((e, E H [0,1].
1.
[px(2, t) (P,(2), t) (P.( +
11111
+ 1
k
Given the boundedness of sp, you (Ix 1) is uniformly bounded.
2. Note that
[lullie
| l k
2u
Yu E IRd VuX
x
1:2
k.a
where X denotes the derivative of X. and that Vax
O if
| lle. > 462. When ||u| S
|ull x2
4
V Va
S
lillo f |ull S l1xll t. Therefore,
R*
(x)
(P
Ip (2x, 1) (p 8]1
X
(Pa
(aa(()(()))))
122
Pa (x))
$
X
(... w. (Px(),
Pa (x)
(Pa(V)
|VV.A(P((w)..)))
+
X
12
00 (I)b(()) - PA + + sp?)
||u|
+ sup uER*
VuX
Pa(v)lle (8)
1.2
RR
<Op (11P)(1)) - Pa (1 + +1t
11-t=s²²)
+ IXxll=0 [Pa Pr(v))lme (1151l +1)
(9)
S(C, + 4|| (11 + 1)) (1{((()))) - -Pa(u)ll + -1-- =()=
<(C, + 4|I (11ella 1)) (10 v1l + + -1--) $
where we apply mean value theorem F$ S and the uniform boundedness - th. in 9
3. Fix (I, 1) E H x [0, 1]. We can find sufficiently large such that || < k. So (x)) <
[ x² = 1. Hence,
and x
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(x, ((x, f), = WH,K (Pa(2)) x(x,t))
Wh, (x)))), - - sx(x,())
I I -ull20 +1((TT(Z), - (xxx)))
1
$ k (x) - =112
Take l + oo and pointwise convergence follows.
Fix (I, E H x [0, 1] and e E (0, z). We can find an integer ky > such that || < kg
2.
Cku
Coko <c
Then Vu E (z), llullx S || c < ko, where (x) is the e-ball around r in H. wk > ko [PP( R22
and Vu (x), X
Therefore,
(2 (2, t) (b.
sup (x)
1u 10 111 Wi (Px()), 1)) (Pk,
sup M.DE (x)
1 111
WA (P)(u),t) Vki,k( Ph (c(()) (P Pk( 1
sup N.DEBM (z)
[Px(u) - (x( RR
|u - v/ lll
1 (Pe(u), t) k/ (P)().t
sup ( (z)
[Pa(u) - (R)
Ju, t) (,,.6(U, t)
sup BY (P)
(10)
| v1llx
JBy (0) Dee (v)( y, - vull y, tdy
sup w,eE (PP
||u - vilRR=
b (u y, +1bx(1' U, t) dy
Tex (w)
sup (P
11(u 1) - (v --y)ll=
B (0)
1 11b u, t)
(bx
sup w.ee w. Bk cto (P)()))
(11)
|uu v1lR
lpp 1) (p(,,))) | - w11x
sup B, (x))
(12)
Lbo(s)) (T (I), +(xE) EK)
Ibp(,1) & (I)) (c), 2c),
(13)
where (10 follows from the fact that || PP((u) - Px $ w|u 211 < c, and holds because
(u y) E B* Biteg (Pe(x)). To justify 12). note that Vu, E B (Px(x)), we have
Σ u.c.
(2)
=|lu- - Pa(x))))) Se,
H
k
U.4 Σ
= ||u - -Vllga
i=1
76
b Σ i=1
Su t), Φ
ujest (x(
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i.e., Vu, E (x)) can pick two points from B (TT( such that the fractions in 11 and (12 achieve the same value. Lastly. (13) follows from our choice ky-
Therefore, wk ko.
(x) S (T)
Let - 0+ and we have
(x) &
Apply upper-semicootinuity of x + L. 6) and we have
sup ()) s sup L (Tx((())))) s (x).))(x))
k
A.3 SUPERPOSITION PRINCIPLES
In this section, we leverage the abstract framework of Theorem 3.4 in Stepanov and Trevisan (017)) to establish the superposition principle in separable Hilbert space. Intuitively, the principle states that any narrolyy continuous curve of probability measures solving the continuity equation can be represented
distribution over paths that solve the associated ODE. In this way, statement about marginal laws is transformed into an ODE statement about sample trajectories. The former specifies the object that neural networks are trained to satisfy implicitly, while the latter specifies the mechanism through which new data are sampled with a trained model. By working directly with curves, we avoid introducing Radon-Nikodym derivative between conditional and marginal laws as required in the FFM framework (Kerrigan et al. 2024). This pathwise view allows us to bypass the absolute-comee continut- assumption inherent to FFM.
A.3.1 FROM CONTINUITY EQUATION TO ODE SOLUTIONS
Let C(O,, 1]: H)) be the space of continuous function from [0, to H. Let E, : E C(O, 1): -
=ff) E H. be the evaluation map, i.e., Ez(z) = z(t). Let (-)+ denote the pushforward. Specifically,
E C[[0, 1]; H) has law 7), i.e, probability measure on the space C(O0, 1]; H), then ((E))
n(( E C(J., 1]; H)) : E2) = 2tt) E A) for a measurable set in H. Recall that 2 [0, 1 - H is
an absolutely continuous curve if and only if is differentiable almost everywhere on 00, 1] and is
in ([0, H)) (Remark 1.1.3 of Ambrosio et al. (008)). We first define the narrow continuity of
measures.
Definition 15. Let {pt} be family of Borel probability measures on a separable Hilbert space H. indexed by f 10. We say that {pr} is narrowly continuous curve if and only if the map t t J f duy continuous for all bounded continuous functions
H. - E.
We say a probability measure rj is concentrated on set A if (AA)
Now we move on to invoke Theorem 3.4 of Stepanov and Trevisan 2017) to develop the superposition principle.
Theorem 16. Let [1a} narrowly continuous curve of Borel probubility measures on the separable Hilbert space H. Let be Borel vector fields such that foo.1 (x]lla(())]l]pdd < co. Suppose (V+, p4) solves the continuity equation in the sense that for all up E A.
d
d |odys = f (Vi,,e)ddp. vi)dpr
(14)
di
There exists a probability measure over C(O0, 1]: H) concentrated on absolutely continuous curves 2: [0 + H such that (Et) 14 for every E (0, 1] and one has
d
J v(((()) (v())(z(1))
(15)
for n-a.ee and a.e. t [0, 1), for all y A.
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Proof. Recall that the operator Vi is defined by (z) (Vp(z), (x))) This directly satisfies condition (3.1) of Stepanno and Trevisan (2017).
Note that constant functions members of A.
By Lemma| 12 we have (Vipp) (x) (Vv((), & [l:(())] which verifies Inequality (3.2) in Stepanov and Trevisan 2017
The assumption f0.11 Sx (2) ddydt < oo corresponds exactly to condition (3.3) of Stepanov and
Trevisan (017).
Condition (3.4) requires that the continuity equation 14 is satisfied by {ue} in the sense of distribu- tions on [0, 1J, which holds by assumption.
Finally, Lemma T4 establishss that assumption A in Stepanov and Trevisan 2017 is satisfied.
With all required conditions verified, we may invoke the superposition principle, Theorem 3,4 of Stepanov and Trevisan (2017), so 7 exists as finite Borel measure. Notice that 1 me(H) 712 1[;0, 2t) E H}) = ((( 1]; H)). So is indeed probability measure.
To apply the superposition principle, we will show that Equation (4) is satisfied by the expected velocity field x and the marginal distributions of the process X,. This will allow us to conclude that 7 corresponds to the law of the induced rectified flow. Before proceeding, reformulate the ODE 15p into more familiar form in the following corollary.
Corollary 17. Let {ur be a narrowly continuous curve of Borel probability measures on the separable Hilbert space H. Let be Borel vector fields such that Sio.1] Sn (z)]ldddd < co. Suppose (Ve, Ht) solves the continuity equation sense that for all yp E A.
d f pddp / (Vip, vi)dp.
There exists a probability measure y over C((0, 1]: H)) concentrated on absolutely continuous curves z: [0,1] - H such that (Ft) = u4 for every E [0, 1] and one has
2tt) z(0) / (=()))
for n-a.e. and Vt [0,1].
Proof. Fix E Ryo. Let XR E C (R) such that x*(s) 1 [s] S and xa(x) = 0 [f] |s| 2 2R. We denote its derivative by AR. Let ge be an orthonormal basis of H.
Fix k. Let (()) = (e((x,e)))( ex). Note that yx.R and (a, exll < R.
V Xa((1, ex))ex + xx((x, ex)) ex)er
By applying Theorem 16 for rpa.e. z and a.e. 1 E [0,1].
d p(,())())) = (VVip.))
dt
If (=((t), ex)] < R, following the chain rule on Hilbert space (Theorem 2.1 of Coleman (2012)) )
d
dt (t,)(2(t)) = (1(t), V(n.(<>(t))
((((),, ex),
(16)
and
(ViP).n)(2(t)) = (VPx,n(2(t)), v.((())) (cx,
(17)
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For each fixed is and R, denote
d R R (2.t) 2tt) doesn't exist (p,.((2(t)) * (Vips.R)
Note N&. has measure 0. Let R,, = n and set No == UK= U Nr. Hence, No. as a countable union of null sets, is also of measure 0 under 7) @ X, where À is the Lebesgue measure
10 1]. Let (2,t) & Nox For each k we can choose n such that R. = n > (2((t), e.)l By (16) and
(77. for rpuee. z and a.e. L [0, 1], (i((), ex) (e (=(()))) for all positive integers k.
So =() = =+(((t)) for g-a.e. 2 and a.e. E [0, 1]. Therefore, for raa.e 2, A E (0,1),((() =
=()) + f/,, (=(ss))sds
A.3.2 VERIFICATION OF ASSUMPTION
Before we move on to the proof of the main theorem, we show that Assumption holds under mild conditions. Recall that + the solution map defined Assumption 2 The law of the random process described by ²()) = (=()(t)) with initial value z00) ~ No is # i.e., for a Borel measurable set B C C(OO, 1]: 72) (o) = (B)).
Proposition 18. Let vx(-) be continuous in an open set 2 C [o, 1] x H. there exists continuous k(-) L' ((0, 1)) such that
[(((z1) (x2))) < k(])== - 22ll. VI E (0,1),21 21,02 E H,
then Assumption holds.
Proof. The existence and uniqueness are implied by Theorem 16.1 and Theorem 16.2 of Pata (019))
Let u, E H and = =u, + Sh v(s, (s))dss, i E 2. Then
-22))))))) S || u2llx + S ((())ll 22(5)
Applying Gronwall's inequality, Vt [0, 1].
[t) 2t(1) S llu vel1x exp (fr k(s)ds)
Hence,
sup [[) - 22(() $ llu uell exp (L. [[M()dds) tE|0,1]
Therefore, the solution map is continuous.
While there is no Lipschitz guarantee for generic neural networks, modern architectures can be designed to satisfy global Lipschitz conditions. Examples include spectral normalization Miyato al. (2018), exact convolution bounds Sedghi al. (2019), and Jacobian regularization Gouk et al. (2021)
A.3.3 THE LAW OF ODE SOLUTIONS
We verify that the law produced by the superposition principle in Corollary 17 coincides with the law of the random process described by &(t) vr (=(()) with initial value 2(0) ~ Plg
Theorem 19. Let v, be Borel vector field on H and consider the absolutely continuous curves 2: [0, 1] H satisfying
=(t)= [( (=(s))) Vt E [0,
(18)
Suppose Assumption hholds for (s, 2) vz) with the solution map denoted by tb and {pu} is a
narrowly continuous curve of Borel probability measures the separable Hilbert space H. If there
exists a probability measure n over C(0,, 1]: H. concentrated on the curves 2 satisfying (18) such
that (Fx) #L pe for every E [0,1). 1), then y to0
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Proof. Let A C(JO, H)) be measurable set.
77( 7 n ((L))
(19)
7( E *(H): 70) E a- ((A)))
n({7 E ((H) : Bo(7) E (A))) (
(20)
~ (A))
(A)
We have (19) and 20
because concentrates on the solutions of the ODE, i.e., is of 0 measure
outside & (H)
A.4 PROOF MAIN THEOREM
Now we're ready to prove the main theorem. Given a stochastic process X, we let x be the expected velocity defined in Definition |
Theorem 20 (Restatement of Theorem 5) Assume X is rectifiable and (Ze) is its induced rectified flow. Then Ze = X₁ in distribution for all e [0.
Proof. Recall that A is the cylindrical functions H. Let Ex be the integral w.r.t. the path measure of process X. Ex, is the expectation w.r.t. the marginal X.. Vyp E A.
d
Ex (xx+x(X++)))) [xx))
Ex [x()((x,]] = lim dt
h-+0
h [p(X+++n) - (X()))
lim Ex
6-+0
h
lim (((++1)) - -sp(Xe)
Ex
(21)
h
d
Ex
dt (Xx)
(V x
Ex
(22)
Ex, BxIx (v (xx()))) x.)
Ex, ( (X,)),)) X2), Exxx [x,}]]
(23)
Ex, ( 5 (Xt): (X., 0)}-
Note that following the mean-value theorem (Corollary 3.2 of Coleman (012 and Assumption 3
(((e4n) (X.)
X1th
X, V
X,
sup .E(O.1]
h
h
So for all h, (e(xca)-p(X)) is bounded above and 21 follows the dominated convergence thcorem
(Theorem E.6. of Cohn (2013)). (22) follows the chain rule (Theorem 2.1 of Coleman (2012)). ). 23
a direct consequence Theorem E. 1l of Cohn (2013).
Let μ4 denote the marginal distribution of X,. Then, d Exlx((x)) = ((Vp(()) vz ()) d I ( So (z, (z,tt),, mb) solves the continuity One the other hand, d Ex[(p(X,,] de
equation in Corollary 17 Hence, there exists a probability measure on C(O 1]: H) concentrated
on absolutely continuous curves such that (Fe) # = μ4 for every i E j0, 1] and
=(1) = =(0) + [ (3,
Theorem 19 guarantees that = 4 # Ho. which the law of the rectified flow (Zz). Therefore, (.,) n p4 implies that Z, X, in distribution.
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A.5 PROBABILITY FLOW ODE
In this section, we review Gaussian measures in separable Hilbert spaces as prerequisite for the proof of Proposition 7
A.5.1 GAUSSIAN MEASURE IN HILBERT SPACE
A linear operator Q on H is called positive if (Qx, 2 0 for all x H. and strictly positive if (Qr, 2² > 0 for all E H {0]. Let { be an orthonormal basis of H. and let Q be positive bounded linear operator. The trace of Q is defined as Tr(Q) - (QSe, Sk), which is independent of the choice of orthonormal basis. self-adjoint, positive, bounded, linear operator Q is called trace class if and only if (T( < 00.
Following Da Prato and Zabezyk 2014), a Borel probability measure js on is called Gaussian for every H and Borel set B C R. the pushforward measure B + ul{z E H (xx h) E B)) is Gaussian measure on R. Any such Gaussian measure on H admits a unique mean vector m E and unique self-ad adj--adjoint positive bounded linear operator Q such that
E Egp u (b, x)] = (m, h), and Egee (h1 - m) (h2, x mm) = (Qh, h2).
for all h, b2, h2 E (Da Prato and Zabezyk 2014). The operator Q is called the covariance operator of Gaussian measure on H. and it is always trace class (see Section 2.3.1 of (Da Prato and Zabczyk 2014)). When Q a covariance operator, there exists a complete orthonormal system & & of eigenvectors of Q with corresponding eigenvalues {Ax ]
(Conway 1990 Na et al. 2025
We define the Martron-Martin space following Pidstrigach al. (2024). For a Gaussian measure (00, Q) on Hilbert space H. with strictly positive Q, the
space Ho is
defined as
Ho (H) with inner product (S.g'Ha = 1Q-1/3 1/2)))
For completeness, we include the definition of an H-valued Q- D-eeer process:
Definition 21. Let Q be trace-class, positive, self-adjoint operator on H. stochastic process W (Wa (Woorr taking values in H is called standard Q-Wiener process if:
1. Wo =0,
2. W has continuous trajectories,
3. W has independent increments,
4. W. - W, (t (t)) all 0 s SSSEST.
A.5.2 PROOF OF PROPOSITION
Lemma 22, Let Q be a strictly positive trace-class covariance operator and k E Rys. Let p be Gaussian measure with mean and strictly positive trace-class covariance operator C =nQ. Then,
P (x)
T
K
Remark 23. Note that although m x E H. does not necessarily belong to Ho. (pr (z),
Ho is still well-defiedd for h Ho. This has been mentioned briefty in Remark of Helin and Burger (2015) and Remark of Lasanen (2012). To see this in more detail. without loss of generality, assume O. Let (G:) be orthonormal basis of H consisting of eigenvectors of Q, with corresponding
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eigenvalues be {2.}. Let N2 be integers and N2 > N1. Then,
N2
(2.Gi)
Ezzp
Ci
Q-21)
Vxa
N+++
N2
(2, Ci.
20
12
=Esups
Q
t0 Ci
N=
=E =Egup
Σ
(2, (er
aQ- Ci S4)
a t
Vti
i=N1+1
N2
(z) Siles
10-th, Ci) 7
(0 -
BRNN
Visyg
i1=N++1
(i) (Q-tthn (Q- 74
Vidy
N1+
N2
Arssnssi Vddg
(Q-bin,
@ e 71
(o-th,,s)) 2
N
Note that Q hh E H. So Ei= ( Ce Cauchy. Hence,
(aRiid
Gin G.-Q-111 Q+11) is
Ei= v2,
Cauchy in (2()). Hence, for z H. (2, 11) = limn (T L2()).
(e.G))
Cin Q 2/1),, well-defined
Vx
Proof. It is sufficient to show
On 0nF(#)dp(z)= J N
-J ((=)((( 2), Hg du(x))
(24)
Recall that { is orthonormal basis of H. consisting of eigenvectors of Q. Let the corresponding eigenvalues be {Ar}. and F is a cylindrical function. Define the projection operator Ps : HHQ +
Rd (x( = (i), (z, (T Let = Px(X) where X ~ (0, C) Then Y ~ (k( = N(m (k) {(K) where (k) E R (R) R m,a) (m, (i), and Σ(k) (CCj. Gi), following
Lemma 2.2.2 of Bogachev (1998) and the characterizations of Gaussian measure on Hilbert space in
Na et al. (2025). We first evaluate the Gâteaux differential 0 F(r):
F(x + 8h) F(x)
F₂F(() = lim 8-50
8
f(Pa(I + 5h)) - f(Px(z))
lim
8-20
8 [(Pu(x) -
lim
8-20
(V))()())), Px(h))
where the last equality follows the property of Gâteaux differential (Chapter 2 appendix, (Coleman 2012
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Hence,
I FnF(())dul Pa(h))
| (Vf(y), p. (g) (Ey PL)
Recall that (R) Zig (OGi (i) = A= and C' is strictly positive. So ( is invertible. By applying
Stein's Identity (Liu and Wang, 2016). we have
Ey B /V [V(()] = [ E (1)))) (( (A)))
Hence,
J H
On OnF(z)du(z) = (Byrntte) Vf(Y)), RM
(1SX Ey [S(Y) - m mm( Pu(li) = Rynet (10))1244) - -m) , Pa (h))
RC
Now that the left-hand side of (24) has been reduced to an expectation of an inner product in Rt, we apply similar procedure to the right-hand side. Let mah = Et (h, Si and h+ == h. Then.
Ex~p
((X) - (0), 1 [
=Ex~p
((X)(x - -m), Txl Hel
+ Exxa (x -m), -)) (t)
It is sufficient to prove
by showing
E ay (1()) (1))),, Pt(hi)) [2_] Exxu F(X)(x m)), Thh)
(25)
and
Ex~μ ((X)(x= -X - -m), h1))))))))) m -0.
(26)
We first show
25)
Let ((k) = (x-1(X X - -m), Twh) Tah) Hence,
s(k) k x=-1 (X m)) Q Tahh)
m
(X m, n 'Q k -
Note that Q-1xgh is well defined as it supported on only a finite number of eigenvectors. S(k) is a Gaussian random variable on IR, by definition of Gaussian measure on H.
Following the definition of the covariance operator Q (Lemma 2.15 Da Prato and Zabezyk 2014)).
(5)( Y, - m,)) Ex<e m, -,,x
m, Gp
(kQQ(=1Q--)mm (mTh,, Ci)xx
(Tah, Si
(h, Sir
Hence, we can apply the conditional mean estimator for joint Gaussian distribution (Theorem 3 of Holt and Nguyen (203))..
(()) Y| = B[s(k) + +[h,, Si (h, GR) (1))) - (k))) = -122())) m (k), Pt(h)) R*
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Note F(X) is measurable by oYY).
Ex-x(F(X)S(*)
manek))) [R] [F(F(X) IIl
By math) ((V)) [r]
Ex
((0) Px Pk(li) )
ml(1)
B
It remains to show 26 1 1-'0-=((- m), Q
Similarly,
define s
(x- -m), )
Then.
(X m. SilH Gis
(h .Sii
8
E
Gi
Vdi
Vhe
HB
(X m, si) .5)
For j S k.
Cov(s", .Y, -
- (b0) Exap
Σ
1 (X m, Gi) xx) m, Si)
A;
i>k
=
(ht Gi)= E xx~m 1[x m, Si) (X m. 5p/25
A;
i>k
(h' (4 (Se, (5
Again, by Theorem 3 Holt and Nguyen (0033).
ElS Y E[S+] + [Cov(S+ Yi mx()) )_.),,...
..... Cov(S- Yv m/ ( m())) 0.
So Ex~ [F()S++ Ey [B+ Y]] Ey EyNEEt (f())E[ YI] =0 Therefore,
[
On OnF()))u(=)= = F(x)(x= (m x), no du(z).
H
We move on to prove Proposition 7
Proof of Proposition 7 Since is bounded, the assumptions of Theorem 7.2 of (Da Prato and
Zabezyk 2014) are satisfied, and therefore the SDE has unique mild solution, which is a random process Y, satisfying
Y = Yo Yo //"' O yy S. -
Vo,aW..
Let M, = exp (1' 3e ds). Let ((e) be orthonormal basis of H. By Ito's lemma (Theorem 4.32 from (Da Prato and Zabezyk 2014)),
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(MY. (MoYo, Sk) [ (MoSi. vondWa) [ (dMay, s + - da
-ev.)
= CYo, + / (Mosar (MoSe.vosdWWa) vondWa)
= Yoo + { MaVardWss Se So)
where the last equality follows Proposition 4.30 from (Da Prato and Zabczyk 2014).
Therefore, Y, - exp - . a, Yo + So exp J o, Vo,dW,.
Following Proposition 4.28 of Da Prato and Zabezyk (0114)
Y, Yo ((((())), k())),
where n(t) =exp( (xxp-= fo ds and x(t) - So exp - Stx o )odr)),ds.
Recall that Mr/2 is the conditional distribution of Yú Yo Following Lemma 22
we have
(y) n(())) PHQ
PeVVo
x(t)
Also,
Y, d 7()){ + va(T)
where U ~ ~N(00,) Q).
Recall that
Y = (()) + vx(U)U where 1o Panta and U N'0 Q) are sampled independently.
Na et al. (0025) aim to solve
min -l
Eyg~Pan Ey vo
s Yi)
Mino Pno
(Yr) dt,
where S: [0, 1] H. to is measurable, and in practice parametrized by neural network. This is equivalent to
M118 /'' Eyg~Pan Ey; Ey: maiin (((,Y)))
(Y:) dl,
min xx *
PPH
because the joint distribution (Y,, Yi
(Y), Y?).
This is minimum mean squared error problem whose solution is the conditional mean
Pany PHO
St
(Y7) Y/
Tn(t)
Y!
| =
x(1)
Y =
Vx(t)
Note that d
k(()
A())
(it) nt), and
#
2/a(1)
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<(t) r/()) ( 17 (5)oods,
s(1) 20(()(() /, 11
+ (/)
oov / 17 (s)oddds +o
o0 (+([))))))
Hence,
Y% :1t) + R(1) U
2/k(t)
= =(-2 th n(()) % + 0. --l1-x(01
x(1] U 2 k/t(()
+ ot U
2 [n(tt Yo + V ut(t
2 rx(f)
2 1 (t(_ U
=-2
v v
Therefore,
1
-r( -
U
Y - -ー=-50X; oX0
- or((
2 vx(l)
So equation 6 is equivalent to
dy, - -1 (Y, S = E|Ý Y = Yijdt.
2
B CONVEX TRANSPORT COSTS AND THE STRAIGHTENING EFFECT
In finte-dimeenionnl spaces, rectified flows are known to reduce transportation costs and exhibit straightening effect, both desirable properties in generative modeling (Liu al. 2022). These properties extend naturally to the setting of the marginal marginal-preserving property established in Theorem The proofs largely mirror those in the miite-dimensional case and involve routine but lengthy bookkeeping. We begin by defining the RectFlow (-) and Rectify() operations.
Hilbert spaces and follow from
Definition 24. Let Xo and X, be H-valued random variables. coupling (1),X)) is called rectifiable if its linear interpolation process X {tX,+ (1 (t) :t [0,1]} is rectifiable. In this case, the 2 = {22 t E [0, 1]} in Equation (2) called the rectified flow of coupling (Xo, X1), denoted Z RectFlow (x), and (Zo, Zn) is called the rectified coupling of (Xu, X1),
denoted as (Zo Z1) = Rectify ((Xu, X1)).
For any convex function c, The following theorem establishes that the Rectify(-) operator does not increase the transport cost Efe( Zo)], which corresponds to special case of Monge's 's formulation of optimal transport (Section 1.2 of Ambrosio et al. (2024)). generalizes Theorem 3.5 from Liu et al. (2022) to the Hilbert space setting.
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Theorem 25. Let Xo. X1 be H-valued random variables such that the pair (Xu, X1) is rectifiable, and define (Zo, Z1) = Rectify( (Xu, X1). Then for any convex function c: H. R.
E E
Zo)l = E -xo)l-
Proof.
E (c 20)l =E E
v (1, Zx) dl
S E
(v (t, Z,) dt
(27)
(1, X())
E
c cEE [[x, - Xo) xxl) dt
E
IE (X1 Xo) |x]
(28)
lc (X1 - -Xo)] dt
E (X Xa)l,
where we apply Jensen's inequality at 27 and 28
Following Liu et al. (022). we define a coupling (Xo, X1) to be straight if it is fixed point of the Rectify(-) operation. Such couplings are practically useful, as their associated process is linear and can be simulated exactly in single step. As direct extension of Theorem 3.6 from Liu et al. (0022) to the Hilbert space setting, the following theorem characterizes this property.
Theorem 26. Let Xo. X1 be H-valued random variables such that the pair (Xo, is rectifiable. Let X, = tX1 + (1 X) and Z RectFlow ((Xo, X1)). The following statements are equivalent.
1. A strictly convex function e: H - RR, such that E [e (Z1 7o)] = Ee Xo))-
2. (Xo, X1) is a fixed point of Rectify()). that is, (Xa,X)) (Zo, Z1).
3. The rectified flow coincides with the linear interpolation process: X Z.
4. The paths of the linear interpolation do not intersect:
[ f Xo E[X: Xo x.]ll dt O.
((Xo, (6)):i))
== (1)):xi))==
Note that V ((Xg, X1)) indicates that X1 X, EX Xo X.] almost surely when
Uniform (10, 11), meaning that the lines passing through each X, is unique, and hence no linear interpolation paths intersect.
Proof. Clearly. 3 = 2 1 1.
4: (Z) %o)] [e - Xo)] = the equalities are achieved in inequalities (7
and 28
Since is strictly convex, (X1 Xx is constant almost surely in X and t e [0.
So X1 X,] = E|X1 - Xol almost surely in X and t [0, 1] and V(X) = 0.
4 3: (X)) 0 = E| XooX,] = E[X1 Xo] almost surely in μr and & E [0, 1]. We
have 10 (X1 - (8) dl E Xa x,) di So nx (1, X.) di for E (0, 1). Hence,
= + 1' (Xi Xx) dt Xo f f'xx (s, X.) Xx) ds.
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By definition, Z also satisfy
Ze = Xo + S oz (s. ds.
By Assumption 2 for any fixed sample w, the solution is unique, i.e,, X(:,) = Z(;w). So X Z.
We next present Hilbert space analogue of Theorem 3.7 from Liu et al. (2022) which shows that repeated application of the rectified flow progressively improves the alignment of the coupling, resulting in more linear trajectories and reduced path overlap. Following [Lu et al. (2022). the straightness of a continuously differentiable process Z = {2+} is measured by by
- (00) [
S(Z) -2)=-Am (1
2.11 2/13gam dr.
Theorem 27. Let Xo. X1 be H-valued random variables such that the pair (Xo; X1) is rectifiable. Let 2h the k-th rectified flow of (Xo, X1), that is, Z RectFlow ((Zb, z{)) and (Zg. 29)
(Xo: X1). Assume each ; 24) is rectifuabe for k = o...., K. Then
K
Σ
S (z + (128, z1)) S E |IXX Xoo1
K=
Hence, if E
[1x; Xoll²]
< +00, we have mingsk K S (2t) + V ((z). z - O(1/K).
Proof.
|
Xool2 = || Xo - E Xo X2]1
+ +(2 No R(X, Xo E[X; Xo x.))
+ | - Xo xell?
Then
E| Xoll² = V((Xo,
B[J|X| XolXxl1
(29)
Note
Zol 2 2 Zi Zo. 2o) + +||z0|?
[Z Zn) 2, 1
By moving the integral over / inside the inner product, we get
S(Z) / 2 di
E1|Z1 - Zoll
(30)
By construction, 2, X[ Xo = Zz]. Therefore, E | = -E - -Xo Xel Hence,
by Equations 29 and
30
E E Xo11² E1 Zoll² = 5(Z) + v((X,,
Applying it to k
E 8[12 18 || 1112/+11220000 +11 -s (z + V 211,
Summing over l leads to the desired inequality.
L
A practical implication of Theorem 27 is that single-step sampling becomes feasible after several recursive applications of Rect Flow (c)
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C EXPERIMENTAL DETAILS
This section provides additional results and implementation details that supplement the main text. We begin with further qualitative results on CelebA to highlight the diversity and fidelity of samples generated by our method. We then outline the architectural configurations and training procedures used in our experiments. The code used for our main experiments is available at https:: //g1 thub com/ 7 Samm/-Jjaaxin/Fuai
ona -eett fied-Flow
C.I. ADDITIONAL QUALITATIVE RESULTS ON CELEBA
To further illustrate the generative quality of the Functional Rectified Flow (FRF) model, we present additional samples. Figures 3 through 7 show images generated using the FRF model trained on CelebA 64 X 64 dataset. These results highlight the model's ability to consistently produce high-quality and diverse samples across multiple runs.
Figure 3: Additional CelebA samples generated by FRF.
C.2 ARCHITECTURAL DETAILS AND HYPERPARAMETER CHOICES
C.2.1 INR ON MNIST
We follow Franzese al. 2023) and adopt the original INR architecture proposed by Sitzmann et al. (2020). The network is fully connected MLP with 8 layers, each containing 128 neurons and using sinusoidal activations (Sitzmann et al. 2020). We implement the modulation-basee meta-learning framework as in Dupont et al. 2022) and Finn et al. (2017). The base network parameters are optimized in the outer loop using the AdaBelief optimizer (Zhuang et al., 2020), with a cosine learning rate schedule ending 10 The inner loop adapts sample-specifie modulation vectors via 3 steps of SGD with learning rate of 10 10-2 Training is performed on 8 NVIDIA A40 GPUs with an effective batch size of 64 8. The model is trained for 106 steps and takes approximately days.
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Figure 4: Additional CelebA samples generated by FRE.
Figure Additional CelebA samples generated by FRF.
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Figure 6: Additional CelebA samples generated by FRE.
Figure Additional CelebA samples generated by FRF.
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C.2.2 TRANSFORMER ON CELEBA
We follow Franzese et al. (2023) and adopt the UViT backbone introduced in Bao et (023).
The model uses 2D sinusoidal positional embeddings, and we set the patch size to effectively
treating each pixel as a token. Our architecture matches that of Franzese al. (2023) consisting of transformer layers, each comprising an 8-head self celf-attention mechanism and fully connected
feedforward layer. Skip connections are employed following both Franzese et al. (2023) and [ao
et al. (202)). For optimization, we use the AdamW W optimizer (Loshchilov and Hutter 2019) with learning rate of 10- The weight decay set to 10-2. Training conducted for 6 x 10 steps with effective batch size of 32, and the learning rate is reduced by a factor of 0.1 during the final 10 steps. The model is trained on 8 NVIDIA A40 GPUs, and the full training takes approximately
cosine warm-up schedule, ending
week.
C.3 NEURAL OPERATOR ON NAVIER-S STOIER DATASET
We follow Kerrigan et al. 2024) and adopt the Fourier Neural Operator (FNO) architecture proposed by Kovachki et (2021) The network comprises Fourier layers with 32 modes and 64 hidden channels, along with 256-dimensional lifting and projection layers, mirroring the configuration used in in/Keriiaan al. (2024). Optimization is performed using the Adam optimizer with an initial learning rate of 5 x 10 which is reduced by factor of 0.1 every 25 epochs. Following Liu et al. (2022), we add small amount of noise to the input data for smoothing. To mitigate redundancy in the dataset, we follow Kerrigan et al. (2024) and subsample 2 x 10a datapoints. The model is trained for 300 epochs with a batch size of 128 on a single NVIDIA A100 MIG device, with the full training taking approximately three hours.
D FAILURE EXAMPLE OF THE FFM ASSUMPTION.
FFM's CM (CM) assumption (errigan et al. 2024) already fails in the fundamental case where X₂ = X1 ~ ~N,0, C) with covariance operator Cen
1 en for an orthonormal basis {en) of L2-
The CM space of C' is He = (m((2/2) = fx Em=1 (I. e// h < oo), where An = 1/n² are the eigenvalues of C. Thus sample x lies in the CM space iff
20 (z,
en, 222 oo.
For X ~ (0, C), the Karhunen-Loèww expansion gives (X, (X,e,) NO0, A,)) so we can write (x,e,) n Sn with En ~ ~N(0, 1) iidd Plugging this into the CM condition yields
(X en
Since c are i.i.d. with Ejg2] 1, the strong law of large numbers implies - e2 = Do almost
surely, so P(X E Hc) O. Consequently, even this trivial setting where no learning is required, draw X, from \/ C) lies in the CM space with probability zero, and the CM assumption imposed by FFM fails in this case.
E EMPIRICAL VERIFICATION OF MARGINAL PRESERVATION THEOREM
To provide an intuitive illustration of Theorem we run synthetic experiment in the Hilbert space (') 1)). We construct a time-dependent process X, that lincarly interpolates between two different random functions: Xo. drawn from a smooth Gaussian process with Matérn-type covariance, and X,, drawn from a mixture of two Gaussian processes with opposite low-frequency mean functions. Using the closed-form expression of the expected velocity associated with this interpolation, we simulate the rectified flow Z, starting from the same distribution at = 0. Figure g shows, for E {0, 0.25, 0.5, 0.75, 1}. shoulder-to-shee to-suuldder plots of 512 sample paths from X,, and Z, as functions on [o, 1). At each time, the cloud of curves generated by the rectified flow is visually indistinguishable from that of the original process, including the emergence and disappearance the bimodal structure.
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F DISCLOSURE OF LLM USAGE
Large Language Models (LLMs) were used solely to assist with polishing the writing and presentation of this paper. All technical contributions were developed entirely by the authors without LLM involvement.
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80
39
62
24
(a) Xo al /
(b) Z, at =
(c) Xe at t = 0.25
(d) Ze at t = 0.25
54-
36
-25
-22
-58
(e) X2 t 0.5
(1) Z1 t 0.5
22
44
10
22
04
54
(g) X, at L 0.75
(h) Zz at = 0.75
-5
-50
(i) X₂ at 1= 1
(i) Z, at 1 1
Figure 8: Shoulder-to-shoulddrr comparison of the original interpolation process X, (left column) and the rectified flow Z, (right column) at different times t E {0, 0.25, 0.75, 1}. Xo is a Gaussian process with Matémnttype covariance and X, is mixture of two Gaussian processes. Each panel displays 512 sample paths in (,2 1]). The close visual agreement between the left and right panels d property 7, X,.
at each time illustrates the
38


Reviews:
[{"review_id": "W0p5uQysbj", "reviewer": "Reviewer_yJzb", "summary": "This paper presents a rigorous theoretical extension of rectified flows to infinite-dimensional Hilbert spaces, establishing a unified framework for functional generative modeling. The authors prove marginal-preserving property for rectified flows in this settings. It is demonstrated that functional mapping of flows and ODE of probabilistic flows arise as special cases within their framework. Empirically, the method achieves state-of-the-art results on images and scientific datasets using three different neural architectures.", "questions": "- The assumption of pathwise differentiability is central to your theory. How is it implemented or tested during training, especially when using discrete data to approximate functions?\n- The paper highlights computational efficiency as a motivation. Could you provide quantitative results (e.g., wall-clock time, number of function evaluations) comparing the sampling speed of your FRF against functional diffusion or flow matching models?\n- In sentence 18, you note that Lipschitz's global conditions can be applied to neural networks. Did you use such methods (e.g., spectral normalization) in your experiments to ensure the correctness of the ODE formulation? If not, did you observe any instabilities during numerical integration?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The Hilbert space extension of rectified flows is novel and rigorous, addressing key limitations of prior functional models by relaxing restrictive measure-theoretic assumptions.\n- The work successfully unifies several functional generative models, providing a cohesive theoretical perspective.\n- Extensive experiments across multiple architectures (INR, Transformer, Neural Operator) and data types demonstrate consistent superiority over competitive baselines.", "weaknesses": "- The theory relies on the assumption that the stochastic process $\\mathbb X$ is \"pathwise continuously differentiable.\" Although this is more plausible than absolute continuity of measures, it is still a non-trivial assumption about the smoothness of data trajectories. It would be useful to briefly discuss the practical validity of this assumption for real functional data (e.g., images as functions, solutions to partial differential equations). Such a statement is not obvious if, for example, the image contains contrasting edges.\n- While the experiments are convincing, they are limited to three datasets. Including a more diverse set of functional data, such as irregularly sampled time series or 3D shapes, would further strengthen the claim of general applicability. Furthermore, a direct comparison of training/sampling speed versus other functional models (FDP, FFM) would be highly informative, given that one of the key motivations for rectified flows is efficiency. Of the three datasets listed, two are datasets with images for which the need to apply this functional technique is not at all obvious -- models built on conventional finite-dimensional principles work well with them.\n- The excellent performance is demonstrated by plugging into established, powerful architectures (INR, Transformer, FNO). An open question is how much of the performance gain is due to the proposed theoretical framework versus the inherent capacity of these architectures. An ablation studying a simpler backbone could help isolate the contribution of the rectified flow objective itself.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OqNU9fTf1I", "reviewer": "Reviewer_JJQ6", "summary": "This paper extends the **Rectified Flow (RF)** framework to **infinite-dimensional Hilbert spaces**, providing a theoretical foundation for functional generative modeling.  \nThe proposed **Functional Rectified Flow (FRF)** generalizes the original rectified flow (Liu et al., 2022) beyond Euclidean domains, offering a deterministic and mathematically rigorous formulation for *function-valued random variables*.  \nOverall, the work aims to lift rectified flow theory into infinite-dimensional settings, positioning it as a unifying framework for functional generative modeling.", "questions": "1. Can the authors provide more intuition on the *practical implications* of Hilbert-space rectification — e.g., does it enable higher-resolution function generation or better memory scaling?  \n2. Could the marginal-preserving theorem be empirically verified (e.g., via sample-wise trajectory interpolation analysis)?  \n3. Is there a computational or convergence benefit of FRF over functional flow matching in terms of training stability or sampling complexity?  \n4. Would FRF remain valid if the underlying Hilbert space is replaced by a Banach space, or if data lies on mixed discrete–continuous domains?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The formulation is mathematically solid, carefully extending continuity equations and rectified flows to Hilbert spaces via the *superposition principle*.  \n2. The work unifies **rectified flow**, **functional flow matching**, and **functional probability flow ODEs** under a single theoretical umbrella, clarifying their relationships.  \n3. Results show modest but consistent improvement in *FID* and *density MSE* metrics compared to functional diffusion processes (FDP) and functional flow matching (FFM).", "weaknesses": "1. The main theoretical advancement—lifting rectified flow to separable Hilbert spaces—is **mathematically correct but conceptually incremental** relative to prior work on functional flow matching (Kerrigan et al., 2024) and functional diffusion models (Franzese et al., 2023). The novelty largely lies in formalizing the *marginal-preserving property* in infinite dimensions rather than introducing fundamentally new generative insights.\n2. Experimental improvements (e.g., FID 0.43 → 0.41 on MNIST) are marginal and within statistical noise.  Experiments are confined to relatively small datasets and fixed architectures; no ablation or analysis of functional-space regularization is included.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "J9ShjqxqcZ", "reviewer": "Reviewer_Yrn2", "summary": "The authors demonstrate theoretically the soundness of most general Rectified Flows on infinite-dimensional spaces, greatly relaxing the assumptions imposed by previous works. They propose three ways to train the proposed method, FRF, which they evaluate on two different modalities, on which they significantly improve on the SOTA.", "questions": "1. To be certain: what are the main assumptions made in this work?\n2. Although they are certainly all essentially equivalent, could you justify formulating this as Rectified Flows (which are not really considered state-of-the-art), instead of, say, Flow Matching [1] or Stochastic Interpolants [2]?\n3. How does this work relate to stochastic processes? Is it the case, similar to finite-dimensional (typically Euclidean) data, that we can recover the same marginals without the Wiener process (probability flow ODE [3])?\n4. Not being an expert on functional ML, how do you train your models on *images*? And why is it a relevant experiment whatsoever? Is there an advantage to this representation over the typical tensorial CxHxW one?\n5. Why did you not try out all applicable methods on the same datasets?\n\nOverall, I think this is a good work, but some clarifications must be made, and mostly the work would benefit a lot from a slightly different presentation, in my opinion.\n\n\n### References\n[1] Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, Matt Le. “Flow Matching for Generative Modeling”\n\n[2] Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden. “Stochastic Interpolants: A Unifying Framework for Flows and Diffusions”\n\n[3] Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, Ben Poole. “Score-Based Generative Modeling through Stochastic Differential Equations”", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The work is well-anchored in the literature regarding the problem it tackles (flows on infinite-dimensional spaces). Good comparisons are offered at the end of section 4.\n- The framework proposed generalises the previous results of the field, and essentially match what we would expect Rectified Flows to be like in this setting; that is to say, they recover essentially the same method, finally, in theory.\n- There are good theoretical discussions in the appendix – namely, about how reasonable the assumptions are in this work. In general, the work in the appendix seems very thorough and introduces the topic well.\n- The proof of the main theorem seems sound, and is well-organised, in general.", "weaknesses": "- On the theoretical part of the paper, there is, I believe, a great problem in the general presentation: as I first read the paper (without the appendix first), I completely missed out on the part about the superposition, and, when I came back to read the abstract, I was surprised to find it there. In general, I would argue that too much of the technical content is hidden in the appendix. Understandably, the content is highly technical and arguably too dense for the main part of a Machine Learning paper. However, essentially omitting main results (included in the abstract itself) seems wrong, and, if space is an issue, other parts (for instance, background) can be trimmed on. I think this is important. Probably some informal/intuitive explanations can be given.\n- While the generality of this method on the interpolation choice and the weaknesses of the other methods are pointed out, it required going through the appendix quite thoroughly to get an idea of what the assumptions made in this work were.\n- While the work is well-grounded with respect to the generative functional literature, it is less so with respect to the generative modelling/flow-based modelling literature, arguably.\n- The empirical validation, I would argue, is the main weakness of this work, and perhaps the method approach in general. It seems that the training objective is pretty standard, so then section 5 must be the novelty, empirically, where different architectures are proposed. Then, in section 6, only one architecture is tried out on each of the datasets. I can imagine that the neural operators can only work in the Navier-Stokes case, but that which worked on MNIST surely can be run on CelebA, unless I am mistaken?\n- Such low FID difference on MNIST, especially at such low FIDs too, may not be used as evidence for improvement. I am not even certain it is a good metric for MNIST. (I agree, though, that it shows that the method works.)\n- I would refrain from using $\\mathcal{O}$ to express “order of” with constants following.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aRimTJnYc4", "reviewer": "Reviewer_CfoQ", "summary": "This paper establishes the rectified flow in Hilbert space, by providing a theoretical result proving the superposition principle in separable Hilbert Space. This justifies the extension of the usual flow matching loss in the Hilbert space,  providing an alternative approach to Kerrigan's Functional Flow matching that not only requires the existsnce of the Radon Nykodim derivative but also the target data to live in the Cameron Martin space of the initial Gaussian Measure.  \nThe paper implements the Flow matching in the Hilbert space by embedding the hilbert space element through functional mapping via implicit neural representation / transformers.", "questions": "Please see the weaknesses section. Does the theoretical soundness of this framework, as a method, provide an advantage over other comparative methods in axis independent from the architectural representation power or way of embedding an element of hilbert space element? \n\nIn a similar note, is there any \"practical\" cases in which X1 lies outside of the Cameron Martin Space. Can you come up with a synthetic case for example to show that the proposed approach is **empirically** superior to functional flow matching? (I guess adding jumps / disconitniuty might do a trick when we set H = L2. )", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "The paper has a solid contribution regarding the superposition principle in separable Hilbert Space, whose significance may even extend beyond the flow matching.  It provides an alternative to the flow matching techniques like Functional Flow matching in the infinite dimensional space.   The paper also provides valuable insights that the proposed theoretical framework can realize the paths/dynamics proposed on other works (Functional Probability Flow ODE, Functional Flow Matching)", "weaknesses": "While the theoretical contributions deserves much credit, but the paper suffers from several weaknesses in its presentations. \n\n### 1.  \nThe theoretical novelty seems orthogonal to the methodological advantage in the current presentation of the paper.  This paper guarantees that \"the flow matching of usual type (Lipman et al, Liu et al)\" is theoretically justified on the Hilbert space.  However, as a method, (4) on its own is not novel, and it is impossible in the current set of results to conclude whether the performance advantage is coming from theoretical soundness of the proposed approach or from the power of the Hilbert-space embedding machinations like Transformer / INR.\nIn order to channel the theoretical novelty to practical significance, I believe an extensive, corner case experiment is required to show that FRM overcomes the problem of FFM; for instance, one may come up with the case in which there is no well defined RN derivative, and show that FRM outperforms FFM that is applied as an an approximate procedure ( with the same representation power allowed to both methods).  Some effort is needed to fundamentally isolate the power of the transformer from the threotical novelty. \n\n\n### 2. \nThe presentation of the novelty of this work was very hard to follow-- the EFM's theoretical requirements are noted, but to what end? I do agree that these requirements are more impractical than the ones required by the proposed approach, but they have not been shown to emerge as the real problem in practice.  So many methods in machine learning are applied to the real world problem with hand-wavy assumptions --- these ambiguities would be of interest in machine learning only if they present a real threat.     Without the presenting that a \"real threat\" is triggered by impractical functional assumptions, the motivation of overcoming EFM's assumptions is weak. \n\nAt the same time, if the goal of this work is purely theoretical,  I feel that EFM deserves different treatment in this paper, as a more restrictive framework purported to the same theoretical goal, for example.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)