On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity
Authors: Quentin Bertrand
Anne Gagneux
Mathurin Massias
Rémi Emonet
Venue: NeurIPS 2025 oral ()
Keywords: flow matching
generalization
memorization
Primary area: deep_learning
TL;DR: We leverage the closed-form formulation of flow matching to understand its generalization

Abstract: Modern deep generative models can now produce high-quality synthetic samples that are often indistinguishable from real training data. A growing body of research aims to understand why recent methods, such as diffusion and flow matching techniques, generalize so effectively. Among the proposed explanations are the inductive biases of deep learning architectures and the stochastic nature of the conditional flow matching loss. In this work, we rule out the noisy nature of the loss as a key factor driving generalization in flow matching.
First, we empirically show that in high-dimensional settings, the stochastic and closed-form versions of the flow matching loss yield nearly equivalent losses. Then, using state-of-the-art flow matching models on standard image datasets, we demonstrate that both variants achieve comparable statistical performance, with the surprising observation that using the closed-form can even improve performance.

On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity
Quentin Bertrand'; Anne Gagneux²; Mathurin Massias Rémi ! Université Jean Monnet Saint-Etienne, CNRS, Institut d' O'Optqque Graduate School, Inria, Laboratoire Hubert Curien UMR 5516, F-42023 Saint- Saint-Etienne, Étienne, France ENNS de Lyon, CNRS, Université Claude Bernard Lyon 1, Inria, LIP UMR 5668, 69342 Lyon Cedex 07, France IInii, ENS de Lyon, CNRS, Université Claude Bernard Lyon 1,
LIP UMR 5668, 69342 Lyon Cedex 07, France *Institut Universitaire de France
Code: https::/github.
Abstract
Modern deep generative models can now produce high-quality synthetic samples that are often indistinguishable from real training data. growing body of research aims to understand why recent methods, such as diffusion and flow matching techniques, generalize so effectively. Among the proposed explanations are the inductive biases of deep learning architectures and the stochastic nature of the conditional flow matching loss. In this work, we rule out the noisy nature of the loss as a key factor driving generalization in flow matching. First, we empirically show that high-dimensional settings, the stochastic and closed-form versions of the flow matching loss yield nearly equivalent losses. Then, using state-of-the-art flow matching models on standard image datasets, we demonstrate that both variants achieve comparable statistical performance, with the surprising observation that using the closed-fome can even improve performance.
1 Introduction
Recent deep generative models, such as diffusion (Sohl-Dickstenn al., 2015; Ho et al., 2020; Song et al., 2021) and flow matching models (Lipman et al., 2023; Albergo and Vanden-Eijnden 2023; Liu et al., 2023), have achieved remarkable success synthesizing realistic data across a wide range of domains. State-of-the-art diffusion and flow matching methods are now capable of producing multi-modal outputs that are virtually indistinguishable from human- -ennarree content, including images (Stability AI, 2023), audio (Borsos et al., 2023), video (Villegas et al., 2022; Brooks et al., 2024), and text (Gong et al., 2023; Xu et al., 2025).
A central question in deep generative modeling concerns the generalization capabilities and underlying mechanisms of these models. Generative models generalization remains a puzzling phenomenon, raising number of challenging and unresolved questions: whether generative models truly generalize still the subject of active debate. On one hand, several studies (Carlini al., 2023; Somepalli et al., 2023b,a; Dar et al., 2023) have shown that large diffusion models are capable of memorizing individual samples from the training set, including licensed photographs, trademarked logos, and sensitive medical data.
On the other hand, Kadkhodaic al. (2024) have empirically demonstrated that while memorization can occur in low-data regimes, diffusion models trained on sufficiently large dataset exhibit clear
Equal contribution. Correspondence: quentin.eerrrandeennn. www.berrrradiiii......
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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signs of generalization. Taken together, recent work points to sharp phase transition between memorization and generalization (Yoon et al., 2023; Zhang al., 2024). Multiple theories have been proposed to explain the puzzling generalization of diffusion and flow matching models. On the one hand, Kadkhodaie et al. (2024); Kamb and Ganguli (2025); Ross et al. (2025) suggested a geometric framework to understand the inductive bias of modern deep convolutional networks on images. On the other hand, Vastola (2025) suggested that generalization is due to the noisy nature of the training loss. In this work, we clearly answer the following question:
Does training on noisy/stochastic targets improve flow matching generalization? If not, what are the main sources of generalization?
Contributions.
We challenge the prevailing belief that generalization in flow matching stems from an inherently noisy loss (Section 3.1). This assumption, largely supported by studies in low-dimensional settings, fails to hold in realistic nigh-dimensional data regimes.
Instead, we observe that generalization in flow matching emerges precisely when the limited- capacity neural network fails approximate the optimal closed-form velocity field (Section 3.2). We identify two critical time intervals, at early and late times, where neural networks fail to approximale the optimal velocity field (Section 3.3). We show that generalization arises predomi- nantly early along flow matching trajectories, aligning with the transition from the stochastic to the deterministic regime of the flow matching objective.
Finally, on standard image datasets (CIFAR-10 10 and CelebA), we show that explicitly regressing against the optimal closed-form velocity field does not impair generalization and can, in some cases, enhance it (Section 4).
The manuscript is organized follows. Section 2 reviews the fundamentals of conditional flow matching and recalls the closed-form of the "optimal" velocity field. Leveraging the closed-form expression of the flow matching velocity field, Section 3 investigates the key sources of generalization in flow matching. In Section 4, we introduce a learning algorithm based on the closed-form formula. Related work discussed in detail in Section 5.
2 Recalls on conditional flow matching
Let Po = (O, Id) be the source distribution" and Pdata the data distribution. We are given n data points (1) (n)) Pdlaaa (8(6) E Rdd The goal of flow matching is to find a velocity field Rdd x [0, 1] + Rdd such that, if one solves on [0, 1] the ordinary differential equation
(0) = no E Rd (t(t) tx))),
then the law of x11 when zo ~ Po is Paata: one says that u transports Po to Pdata- For every value of between 0 and 1, the law of zff) defines probability path, denoted p(-It) that progressively transforms Po to Pdata- If one knows the velocity field u, new samples can then be generated by sampling zo from po, solving the ordinary differential equation, and using x11) as the generated point.
In conditional flow matching, finding such a velocity field u achieved in the following way.
(i) First, define a conditioning variable 2 independent t, e.g., 2 = I1 ~ Pdata
(ii) Then, chose a conditional probability path p(-|z, (2, !), e.g. p(-1 = 1,, () = N1(o,, (1 Id).
Through the continuity equation (Lipman et al., 2024, Sec. 3,5), the choice (ii) of the conditional probability path p(+]z,t) defines conditional velocity field cond (xr, 2, t). With the choices (i) and (ii), the conditional velocity field writes
cond (x, 2 x1,
T1
(2)
tt choice Pa = _V Id) is made for simplicity; more generic choices are possible and the reader can refer Lipman al. (2024); Gagneux et al. (2025); Gao et al. (2025) for deeper introductions to flow matching.

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The choice (ii) of the conditional probability paths p(-1z 21,t) t) fully defines probability path p(-lt) (by marginalization against 2) and thus defines an optimal velocity field u" (through the continuity equation), that transports po to Pdaat. (Lipman et al., 2023, Thm. 1)
u' (x, t) Esja,e u cond (x, 2,
(3)
Hence, the optimal velocity u* could be approximated by a neural network ug : IRd x [0, 1] IR. with parameters 0 by minimizing
(0) = E (uppuu - (x 1)
ms--p(-()
However, u" usually (believed) intractable, as a remedy, Lipman et (2023, Thm. 2) showed that (0) is equal, up to a constant, to the conditional flow matching loss. With the choices (i) and (ii)
made above, the conditional flow matching loss reads
(f) = EE, zar-po (Te, () - 11 conodd [It, 2 = T1, 11?
(5)
21 pppaaa (0011
=21 -0
where T4. := (1 t) + tr The objective LCFM is easy to approximate, since is easy to sample from p0 .' (0, Id) and (((o, 1)); sampling from Pdata is approximated by sampling from Pdata = Σ i=1 6z(0)- Although seems natural, replacing Pdata by Pdata in (5) has very important consequence: it makes the minimizer '' of LPM available in closed-form, which we recall below.
Proposition (Closed-form Formula of the Optimal Velocity), When Pdata is replaced by Pdata- with the previous choices (i) and (ïi,, the optimal velocity field i (3) has a closed-form formula:
(1 t) x(1)
(x, = Σ
-
||I ( 2(1-r)² )==I......)
with ((r,t) softmax((
E I
The notation ü* emphasizes the velocity field is optimal for the empirical probability distribution Pdata. not the true one Pdata- Since nondd (xt = ,((() 1) xx f/(6) v, the optimal velocity field a" is weighted average of the n different directions ( - x. Note that the closed-form formula in Equation (6) can be found in various previous works, e.g., Kamb and Ganguli (2025, Eq. 3), Biroli et al. (2024), Gao and Li (2024), Li et al. (2024) or Scarvelis et al. (2025), and can be generalized to other choices of continuous distribution Po (e.g., the uniform distribution, see Appendix A.
From Equation (6), as t + 1, the velocity field ût diverges at any point x that does not coincide with one of the training samples g(((), and points in the direction of the nearest (:())_ This creates paradox: solving the ordinary differential equation (1) with the velocity field û* can only produce training samples (0( (see Gao and Li 2024, Thm. 4.6 for a formal proof). Therefore, in practice, exactly minimizing the conditional flow matching loss would result in Ug û* meaning the model memorizes the training data and fails to generalize. This naturally yields the following question:
How can flow matching generalize if the optimal velocity field only generates training samples?
Investigating the key sources of generalization
In this section, we investigate the key sources of flow matching generalization using the closed-form formula of its velocity field. First in Section 3.1 we challenge the claim that generalization stems from the stochastic approximation coud of the optimal velocity field i*. Then, in Section 3.2 we show that generalization arises when uy fails to approximate the perfect velocity ät. Interestingly. the target velocity estimation particularly fails at two critical time intervals. Section 3.3 shows that one of these critical times is particularly important for generalization.
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Figure 1: We challenge the hypothesis that target stochasticity plays major role in flow matching generalization. Figure la, the histograms of the cosine similarities between â" ((1 - t)my + tæ t) and cond ((1 1) + 1x1, = x1,1) 1) = I1 xn are displayed for various time values and two datasets. For real, high-dimensional data, non-stochuxticity arises very early (before /: = 0.2 for CIFAR-10 with dimension (3, (..2232), 32)). Figure Ie displays the alignment between â" and u cond over time for varying image dimensions d on Imagenette.
3.1 Target stochasticity is not what you need
One recent hypothesis that generalization arises from the fact that the regression target coud of conditional flow matching is only a stochastic estimate of ût The fact that the target regression objective only equals the true objective on average is referred to by Vastola (2025) as "generalization through variance". To challenge this assumption, we leverage Proposition 1, which states that the optimal velocity field â" (x, t) a weighted sum of the n values u (z, i, z = x(()) =-==
for E [n]. and show that, after a small time value t, this average is in practice equal to a single value in the expectation (see Figures la and 1b).
Comments on Figure 1a. produce Figure 1a, we sample 256 pairs (æo, xi) from Po X Pdata- For each value of f,, we compute the cosine similarity between the optimal velocity field ât ((1 t)zg + tz,,t) t) and the conditional target u gond ((1 t)æg + tæ 2 Z1,t)
Zg. The
resulting similarities are aggregated and shown as histograms. The top row displays the results for the wwo-moons toy dataset (d = 2), and the bottom row displays the results for the CIFAR-10 dataset (Krizhevsky and Hinton 2009, become increasingly concentrated around 1, indicating that ä aligns closely with single conditional vector cond From Equation (6), this corresponds
3072); 50k for both. As increases, the histograms
collapse towards of all but one of the softmax weights (, t). This time corresponds to the collapse time studied by Biroli et al. (2024) for diffusion; we discuss the connection in the related works (Section 5). On the two-moons toy dataset, this transition occurs for intermediate-to-lle la-ge values of t, echoing the observations made in low-dimensional settings by Vastola (2025, Figure 1). In contrast, for high-dimensional real datasets, i* (1r, 1) aligns with single conditional velocity field (6) x, even at early time steps, suggesting that the non-stochastic regime dominates most of the generative process. This key difference between low- and high-dimensional data suggests that the transition time between the stochastic and non-stochastic regimes is strongly influenced by the dimensionality of the data.

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Figure 2: Failure to learn the optimal velocity field, CIFAR-10. Left: The leftmost figure represents the average error between the optimal empirical velocity field i" and the learned velocity ue for multiple values of time t. Middle: The middle figure displays the FID-10k computed dataset, using the DINOv2 embedding. Right: The rightmost figure displays the average distance between the generated samples and their closest image from the training set for reference, the horizontal dashed line indicates the mean distance between an image of CIFAR-10 train and its nearest neighbor in the dataset. All the quantities computed/leamed varying number of training samples (10 to 10*) of the CIFAR-10 dataset.
the test
Comments on Figure 1c. To further illustrate the strong impact of dimensionality. Figure Ic reports the proportion of samples 24 (from batch of 256) for which the cosine similarity between û' and gond x (9) exceeds 0.9, function of time f. This analysis is performed across multiple spatial resolutions the Imagenette dataset (Howard, 2019), obtaining dim x dim images by spatial subsampling. Figure Ic reveals sharp transition: the dimensionality increases, the proportion of high-cosine matches rapidly converges to 100%. A practical implication of this behavior is that, for sufficiently large t, if 2n ~ Po and (1) ~ Paatas then û ((1 - t)æ + tx,(0) t) is approximately proportional to x()) - x. Consequently, regressing on (f) or on the conditional velocity 21 xo becomes effectively equivalent. Section 4 investigates how to learn regressing against optimal velocity
field ū*, and empirically shows similar results between stochastic and non-stochastic targets.
The regime where flow matching matches stochasticity is mostly concentrated on very short time interval, for small values of f. We hypothesize that the phenomenon observed here on the optimal velocity field d' has major implications the learned flow matching model ug, which we further inspect in the next section.
3.2 Failure learn the optimal velocity field
This subsection investigates how well the learned velocity field approximates the optimal/ideal velocity field ū* and how the quality of this approximation correlates with generalization. To do so, we propose the following experiment.
Set up of Figure 2. To build Figure 2, subsampled the CIFAR-10 dataset from 10 10 samples. For each size, we trained a flow matching model using a standard 34 million million-parametts parameter U-Net (see
Appendix D for details). Following Kadkhodaie al. (2024), the number of parameters of the network wo remains fixed across dataset sizes. Importantly, the optimal velocity field n* itself depends on the dataset size: the number of samples increases, the complexity of û' also grows. Thus, expect the network ug to accurately approximate the optimal velocity field â* for smaller dataset sizes.
Comments on Figure 2. The leftmost plot shows the average training error
E morpo 3i
[| (xe, () a" (xx, 1), where : = (1 ()z0 + tx
between the learned velocity up and the optimal empirical velocity field i*, evaluated across multiple time values and dataset sizes. With only 10 samples (darkest curve), the network uo closely approximates it. As the dataset size increases, the complexity of ût grows, and the approximation by ug becomes less accurate. In particular, the approximation fails at two specific time intervals: around
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Figure 3: Generalization occurs at small times CIFAR-10 (left) and CelebA 64 (right). Top: Generalization (distance between generated samples and training data) of hybrid models that follow d" [0, r]. then ug [T. 1]. The four colored curves correspond to four specific zo. the black dashed curve is the mean distance over the 256 generated images. Bottom: visualization of generated images for the four different starting noises and various values of (the background color matching the curve in the top figure). Following üt* until 7 2 0.3 yields a model that is not able to generalize.
1 0.15 and near / 1. The failure near is expected, as û* becomes non-Lipschitz at Interestingly, the early-time failure at / 2 0.15 corresponds to the regime where ü* and uound start to correlate (see Figure
Section 3.1). The middle plot of Figure 2 reports the FID-10K, computed on the test set in the DINOv2 embedding space (Oquab et al., 2024), for various dataset sizes. For a small dataset (e.g., #samples 10), ug approximates i* well but does not generalize the test FID exceeds 10S. As the dataset size increases (1000 < #samples < 3000), the approximation ug becomes less accurate. Despite this, the model achieves lower FID scores on the test set but still memorizes the training data. The rightmost plot of Figure 2 illustrates this memorization by showing the average distance between each generated sample and its nearest neighbor in the training set. For larger datasets (#samples 3000), this distance increases substantially, indicating that the model generalizes better. Overall, Figure also suggests that the FID metric can be misleading, even when computed the test For example, the model trained with 1000 samples has low test FID but memorizes training examples.
Figure 2 confirms that generalization arises when the network up fails to estimate the optimal velocity field 2", and that this failure occurs at two specific time intervals. Section 3.3, we investigate which of these two intervals is responsible for driving generalization.
3.3
When does generalization arise?
To investigate whether the failure to approximate ü* matters the most at small or large values of t, we carry out the following experiment.
Set up of Figure We first learn velocity field us using standard conditional flow matching (see Appendix D), then we construct hybrid model: we define a piecewise trajectory where the flow governed by the optimal velocity field i* for times [0, r). and by the learned velocity field ug for times t [T. 1], for a given threshold parameter E [0. 1]. For the extreme case = 1, the full trajectory follows û*,' and samples exactly match training data points. Conversely, when 7 = the entire trajectory is governed by ne, yielding novel samples. Intermediate values of 7 produce a mixture of both behaviors, which we interpret as reflecting varying degrees of generalization. To assess generalization, we measure the distance of generated samples to the dataset using the LPIPS

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metric (Zhang et al., 2018), which computes the feature distance between two images via some pretrained classification network. We define the distance of generated sample x' to a dataset D {x(2), ....,, } as dist(æ; D) ming(eeD LPIPS(2, = We fix a random batch of 256 pure noise images from po- Then, for various threshold values T, we generate 256 images with the hybrid model, always starting from this batch. Finally, we measure the creativity of the hybrid model as the mean of the aforementioned LPIPS distances between the 256 generated samples and the dataset.
varies, the CIFAR-10 10 (left) and CelebA - 64 x 64 (right) datasets. For T < 0.2, the hybrid model remains as creative as vo, despite following ú* in the first steps. For T 0.2, the L.PPS distance starts dropping. On the displayed generated samples (bottom rows), we in fact see that as soon as r 2 0.4, the sample generated by the hybrid model is almost the same the one obtained with ä* (r = 1). This means the final image is already determined at = 0.4, and despite the generalization capacity of the learned velocity field ug. following it only after 2 0.4 is not enough create new image: generalization occurs early and cannot fully be explained by the failure to correctly approximate u² at large t.
Comments on Figure 3. The top row displays the LPIPS distances
Although we have shown that the stochastic phase was limited to small values of in real-data settings, we have not yet definitively ruled it out as the cause of generalization. In the following Section 4, we introduce a learning procedure designed to address this question directly.
4 Learning with the closed-form formula
In this section, in order to discard the impact of stochastic target on the generalization, we propose to directly regress against the closed-form formula in Equation (6).
4.1 Empirical flow matching
Regressing against the closed-form ût, defined in Equation (6), a point (æ4, t) requires computing weighted sum of the conditional velocity fields over all the training points x(0), For dataset of n samples of size d, and a batch of size [B), computing the weights of the exact folosed-frmm formula ii (x, t) of flow matching requires On x [B] x d). These computations are prohibitive since they must be performed for each batch. One natural idea is to estimate the closed-form formula û' (Equation (6)), by a Monte Carlo approximation (Equation (8)), using M S n samples 6(1),..., (MM)
(0) E
||ua(Er,)) t) 0hg (z2, t)l
21 Ddara [0.1] 6(2) ((05) (100)paaaa Paata
with x4 = (1 t)am 1x,,, ((1) = 2T, and
M
bD) - x
((-112 -txxtxx001 11) )
(x, = Σ (xx
1-t A(I,t) = softmax -oftmax
(8)
2(1 - +)2
The formulation in Equation (7) may appear naive at first glance. Still, it hinges on a crucial trick: the Monte Carlo estimate is computed using a batch that systematically includes the point 31, that generated the current zy. If instead ((1) were sampled independently from Pdata. this could introduce a sampling bias (see Ryzhakov et al. 2024, Appendix B, and the corresponding OpenReview comments' for an in-depth discussion). Proposition 2 shows that the estimate úz, is unbiased and has lower variance than the standard conditional flow matching target.
Proposition 2. We denote the conditional probability distribution p(z - (0)) | xx, E) over (x(0)) by (2 1.))) With no constraints on the learned velocity field ua.
i) The minimizer of Equation (7) writes, for all (x,,)
E g(t) (paaa [arr(x, 1)
((2) B waraaa
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ii) In addition, for all (z, t). the minimizer of Equation (7) equals the optimal velocity field, i.e.,
E P (1r, () = -i (x.1).
(10)
6:4) .....b BLM) Bansn
ii) The conditional variance of the estimator Uh is smaller than the usual conditional variance:
Vargtss [ cond (x, b00
Var 8(8) ( (62) y.... (LM) MaM)aaaat Paata
(11)
The proof of Proposition 2 is provided in Appendix B.3. The estimator S5, of the optimal field i* is closely related to self- noff-normalized importance sampling (see Appendix B.2 and Owen 2013, Chap. 9.2), as well as to Pho--Baacwelliinggg 2022). As discussed in Ryzhakov et al. (2024), self-normalized importance sampling estimators of 8" are generally biased, in the sense that: Escr,... M.M) "hr (x2, * (z), A key insight is that our estimator includes ((1) ~ P.atal- x2, t), which leads to the main result of Proposition 2. In Section 4.2, we demonstrate that Algorithm 2, designed to solve Equation (7), yields consistent improvements on high-dimensional datasets such as CIFAR-10 and CelebA. Additional details on the unbiasedness of LEFM can be found in the supplementary material (Appendix B). From a computational perspective, despite requiring M additional samples, Algorithm 2 remains significantly more efficient than increasing the batch size by a factor of M: the M samples are merely averaged (with weights), while the backpropagation remains identical to that of Algorithm 1.
estimators (Casella and Robert, 1996; Cardoso et al.,
Algorithm 1 Vanilla Flow Matching
Algorithm Empirical Flow Matching
for k in 1, ...., Miter do
param: : M // Number of samples in the empirical mean for k in 1, 1...., Vitter do
t ~ U([0. 1))
To NN. Id), ~ Pdatas
To ~ NN0,, Id), I1 ~ pdaata f. ~ u([0, 1])
Ft- = (1 11-t) -)xx0 + 1x
Iq (1 t)aro + tæ1
cond (Izz t) =
31 ftt
8(1) 2
1 1-t
= 11 To
Vj [2 663) ~ Pdata // Samples from Pdata
[(0) = || |eo (zt, t) - u wonnd (2 1|
03 (ze, () = Σタ 66---
softmax -- }} 2(1-1)²
Compute ((() and update
1-1
return uo
((0) = 1| (1x, t) - u (Iz, 1)
Compute (0) and update
return Ugg
4.2 Experiments
We now learn with empirical flow matching (EFM, Equation (7) and Algorithm 2) in practical high-dimensional settings. Our goal with this empirical investigation is first to observe regressing against a more deterministic target leads to performance improvemen//degradationn
Datasets and Models. We perform experiments on the image datasets CIFAR-10 (Krizhevsky and Hinton, 2009) and CelebA 64 x 64 (Liu et al., 2015). For the experiments, we compare vanilla conditional flow matching (Lipman et al., 2023; Liu et al., 2023; Albergo and Vanden-Eijnden 2023), optimal transport flow matching (Pooladian et al., 2023; Tong et al., 2024), and the empirical flow matching in Algorithm 2, for multiple numbers of samples M to estimate the empirical mean. Training details are in Appendix D.
Metrics. To assess generalization performance, we use the standard Fréchet Inception Distance (Heusel et al., 2017) with Inception-Ve V3 (Szegedy et al., 2016) but we also follow the recommendation of Stein al. (2023) using the DINOv2 embedding (Oquab et al., 2023), which is known expressive and discriminative embedding, that leads to less biased evaluation. We also measure the FID between the generated and the train and test sets, rather than only on the training set, as is often done in generative modeling benchmarks. On Figure 2, we also displayed a memorization metric that would detect a pure copy of the training set. Overall, defining and quantifying the generalization ability of generative models is overall a challenging task: train and test FID are known to be imperfect (Stein et al., 2023; Jiralerspong et al., 2023; Parmar et al., 2022), yet no superior competitor has emerged.
more
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Figure 4: FID computed on the training set (50k) and the test set (10k) using multiple embeddings, Inception and DINOv2. Regressing against a more deterministic target (EFM 128, 256, 1000) does not yield performance decreases. On the contrary, the more deterministic the target, the better the performance.
Comments Figure Figure 4 compares vanilla flow matching, OTCFM, and the empirical flow matching (EFM, Algorithm 2) approaches using various numbers of samples to estimate the empirical mean, M E (128, 256, 1000]. First, observe that learning with more deterministic target does not degrade either training or testing performance, across both types of embeddings. On the contrary. we consistently observe modest but steady improvements as stochasticity is reduced. For both CIFAR-I0 and CelebA, increasing the number of samples M used to compute the empirical mean-i.e,, making the targets less stochastic-leads to more stable improvements.
worth noting that Algorithm 2 has computational complexity of O(M x [B] x d), where [B] is the batch size, M is the number samples used to estimate the empirical mean, and is the sample dimension. In our experiments, choosing M = [6] = 128 yielded a modest time overhead. For empirical flow matching, we experimented with several values beyond M = 1000 (e.g., g. M 2000, M 5000). The results were nearly identical to those obtained with M 1000, with curves being visually indistinguishabl.. Therefore, we chose not to report results for M 2 1000.
5 Related work
The existing literature related to our study can be roughly divided into three approaches: leveraging the closed-form, studies on the memorization vs generalization, and characterization of the different phases of the generating dynamics.
Leveraging the closed-form. Proposition 1 has been leveraged in several ways. The closest existing work is by Ryzhakov et al. (2024), who propose to regress against â* as we do in Section 4. Nevertheless, their motivation is that reducing the variance of the velocity field estimation makes learning more accurate: as explained based intuitions (e.g., Figure 1, challenged by Section 3.1). The idea of regressing against a more deterministic target (as Proposition 2 shows) derived from the optimal closed-form velocity field has also been empirically explored for diffusion models (Xu al., 2023). Scarvelis al. (2025) bypass training, and suggest using a smoothed version of û" to generate novel samples. In work specific to images and convolutional neural networks, Kamb and Ganguli (2025) suggested that flow matching indeed ends up learning an optimal velocity, but that instead of memorizing training samples, the velocity memorizes combination all possible patches in an image and across the images. They show remarkable agreement between their theory and the trajectories followed by learned vector fields, but their work is limited to convolutional architectures, and was recently extended larger class of architectures (Lukoianov et al., 2025).
Section 3.1, we argue this claim rests on misleading 2D-

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Memorization and reasons for generalization. Kadkhodaie et al. (2024) directly relates the transition from memorization to generalization the size of the training dataset, and proposes geometric interpretation. We provide a complementary experiment in Section 3.2, quantifying how much the network fails to estimate the optimal velocity field. Gu et al. (2025) provide a detailed experimental investigation into the potential causes for generalization, primarily based on the characteristics of the dataset and choices for training and model. Vastola (2025) explores different factors of generalization in the case of diffusion, with special focus the stochasticity of the target objective in the learning problem. Through a physic-based modeling of the generative dynamics, they study the covariance matrix of the noisy estimation of the exact score. In our work, we believe that have shown that this claim was not valid for real nigh-dimensiona data. Niedoba et al. (2025) study the poor approximation of the exact score by the learned models: like Kamb and Ganguli (2025), they suggest that the generalization of the learned models comes from memorization of many patches in the training data.
Temporal regimes. Biroli et al. (2024); Sclocchi et al. (2025) provide an analysis of the exact score, the counterpart of the exact velocity field for diffusion. For multimodal target distribution, the authors identify three phases (we keep the convention that = is noise and t < t1, all trajectories are indistinguishable; for t < t < 12, trajectories converging to different modes separate; for t > f2. trajectories all point to the training dataset. In the case of Gaussians mixtures target. they highlight the dependency of in the dimension and the number of samples, in O ((log n) /d),, meaning that the first phases are observable only if the number of training points exponential in the dimension. The methodology they adopt to validate the existence of such t2 on real data relies on the stochasticity of the backward generative process, which does not hold in the case of flow matching. Our experiments learned flow matching models allow us take this theoretical study on memorization and temporal behaviors of generative processes a step further.
is target): for
Conclusion, limitations and broader impact
Conclusion. By challenging the assumption that stochasticity in the loss function is key driver of generalization, our findings help clarify the role of approximation of the exact velocity field in flow matching models. Beyond the different temporal phases in the generation process that we have iden- tified, we expect further results to be obtained by uncovering new properties of the true velocity field.
Limitation. Our work is mainly empirical, with focus on learned models, but did not precisely characterize the learned velocity field, in particular, how it behaves outside the trajectories defined by the optimal velocity. Leveraging existing work on the inductive biases of the architectures hand seems like promising venue. Another limitation is that we did not investigate the interaction between the architectural inductive bias, and optimization procedures: this is a very challenging, but active area of research (Boursier and Flammarion, 2025; Bonnaire et al., 2025; Favero et al., 2025).
Broader impact. We hope that identifying the key factors of generalization will lead to improved training efficiency. However, generative models also raise concerns related to misinformation (notably deepfakes), data privacy, and potential misuse in generating synthetic but realistic content.
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7 Acknowledgments
The authors thank the Blaise Pascal Center for its computational support, using the SIDUS (Quemener and Corvellec, 2013) solution.
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20

---

A Proofs of Section 2
p(a/ 2 = 2(().t) .t)
i (2z 1) =
u cond (x, (x, =====,,,1)
(12)
Σx= p(z|z = x((1) .)
i=1
A.1 Proof of Proposition 1
Proof. In the case where Pdasta conditional probability writes
p(e,6,==att x7
p(= xz (z,,t)
(13)
p(s,t)
platt,2 2= 111)) )pll, 2 z(0))
(14)
Mx;11 1)
plalt, 2 = x110 ppttz = x(0))
(15)
Σx= 1(15, 1, = x((())
pzll, 2 = xl0h) pp= = .0) Σx p(=lt, = = x(()) p(= = )())
(16)
p(ztt, x- x1)
(17)
- p(zlt, = = x (t))
Pluging Equation (17) in Equation (3) yields the closed- formed formula for the velocity field:
2n (z, t)
u rond
(I,t, = )((= x 12/1
(18)
plzlt, 2 = x1) Σ =1 p(z|112 2 = x(()
cond (I, t,, = (1)
(19)
which proves Equation (12); using that xlt, = z(1) ~ N (tz (1 t)² Id) and u uond tr, t, =
x((()= = -(ا) I-t yields Equation (6).
For the case 2 Po x Pdata.
at (s, 1)
cond (2r, l, 2) 2)p(2/1r,)) () dz
(20)
oond (I, t, p(x, 2,t) dz p(x, 1)
(21)
p(x| l2 t)p(2))(t))
cond (I, 1, x)
J. p(ar 2') ) (p)(()())))))))))))))))))))))( da de
(22)
u rond (xr, 2)
[g platt, 2'))(=")) dz dz
(23)
Since 2 ~ Po x Pdntas the denominator is equal to:
1. p(a)t, 2')x(²)) ((()) dz' = / Σ 5z (a1 - -t)a0 + (1)
11
exp( 1
2 -0)) =g)
(24)
v (2
i=1
n
(v(
(2T)) d exp(
2(1 - t)2 1ly (11)
1
(1 - t)a dy
1
(y = (1
-t)x + (2)
n y
(25)
exp
2(1 t)2 11 - tx (1))
1
(26)
/2TT 1 - -jjyd
21

---

Likewise, the numerator equals:
|
(x 20)02 1)20 + 1x(
exp(
2 1 12)
(2
==1
(27)
(6) - (z(
exp(
11
to (6) 112dy
11
-
((2 (2)
2(1 - 1)2
(28)
(6)
(-2(1
1
1 to (x) )12
exp
(2 - t)3ja
2(1 t)2
(29)
Taking the ratio of Equations (24) and (29) concludes the proof.
B Additional details and comments on empirical flow matching
First, recalls on the optimal velocity (Equation (6)) and the empirical flow matching loss (Equa- tions (7) and (8) are provided in Appendix B.1. The unbiasedness of the estimator presented in Appendix B.2, and its proof is in Appendix B.3.
B.1 Recalls
The closed-form formula of the "optimal" velocity field is:
(1)
softmax ((-1ss7--)
|1 1111--100k) 1o0( 2(1 t)2
i (x.
(6)
k=
The proposed loss uses mini-batches of size M (instead of all n training points) to build an estimator is of i*:
LErM (0) =
t~11{[0
[uo(xr,)) () - u5h (Zt, 1)]
2g -Po
Paaat
31 (1- () tan
(()) = 612)
with
M Σ j=1
b(t) It.
)
((-ley -6(())12
05 (Zz, t)
=
softmax
(8)
1
t
2(1 - t)2
k=1....M M
Crucially, Equation (7) the sample b( depends on 2, and reused in the estimate üās. This important detail yields an unbiased estimator of it..
B.2 Theoretical properties of the proposed estimator
First, discuss below the relation between Proposition and the sampling literature.
Links with importance sampling. The estimator û* in Equation (6) can be seen as form of importance sampling (see Robert et al. 1999, Chap. 3 for an in-depth reference). In nutshell, importance sampling is a way to estimate an expectation when one cannot easily sample from the random variable it depends on. More precisely, in the ideal case the velocity field formula is the following
Pdata (as opposed to Pdata),
(xz, t) EEjrett [u cond (zr,
(30)
u cond (æt, z, (0)p(pp((2.e)) t)dz
(31)
22

---

When 2 ~ Pdata it is difficult to sample from 2/ t, but the latter equation can be rewritten as
u² (Iz: t) = -. cond (zt, 2, t) p(z t)
u
p(z)dz
(32)
p(z)
and one can easily sample from 2 ~ Pdata using the empirical data distribution z(1), ..... (())
cond (Iz, (1) p(2 z11) (xe,t)
u" (
(33)
plx(())
coud (I (0) ,1((== = p21,))
(34)
1=1
= ú" (zı,t)
(35)
B.3 Proof of Proposition
We first recall Appendix B, which we prove in this section.
Proposition 2. We denote the conditional probability distribution p(2 = x(0) by Pdora xr, With constraints the learned velocity field uo.
| x, t) over {x
The minimizer of Equation (7) writes, for all (x,
E
(ea (-11,8)
(iM (2r, ()]
(9)
6(0) M) Popaa
ii) In addition, for all (zz t). the minimizer of Equation (7) equals the optimal velocity field, i.e.,
E D0 (ppaaaaantt (-1.t) [w' (2r, () = û" (x,
(10)
( ........B..h
tii) The conditional variance of the estimator Th is smaller than the usual conditional variance:
Var 60) peree (12,1) (x, [)], S Vargo) (a cood (x, b()) .t]
(11)
( b
B Baata
Proof of Item (i)) With no constraints on ug, the empirical flow matching loss writes:
E
[ t) ais (Zx,, t)]
(36)
=1=(1 (1-1) --f00tottt
60)= : (29) ..... 6... MM Pasaa
E.- Egess Aaaaa Ma (1)))) 2prept
I (2te t) 0i5 (xte, t)ll
(37)
((2) ..... 6080) lacs
-E y (0 Pama (122.)) ||uo(Er,)) 1) -is (2t, 1)1 (00)) udaata
because ((2) ..... ((ad) IL 1t, (38)
which is minimized when for all Weg
(xe, f) E (100 Maaia (1a0)) 812) ((0)) Badat
[iAs (Iz, t)]
(39)
Proof of Item (ii). The minimizer for given (æ,, t), removing these elements from the notation for conciseness and abstraction, is a weighted mean:
a* (xt, 1) = =ū= Σ w (
with
(40)
1=1
n
() Paaata (2 1011 (t,)(tzz)) xx)
Σ
(
(41)
() coud (22, (0.)..
(42)
23

---

We express mini-batch as an M-valued vector of indices, i E [[1, m2] from Equation (7), considering the definition of the softmax, can be expressed as mini-bate weighted-mean:
M. The mini matic estimate
Σ dd ((())(()) (1)
(43)
Σ Ej= j= w(it)
The categorical distribution over [1, ni] with probabilities following the weights w in (41) is denoted (al(w) and the uniform distribution, i.e., Cat(1/n)), denoted Unif.
The main result the following that, in expectation over the biased- mini-miniiaa bati-b where the first point is drawn according to and the M 1 other points drawn uniformly, the mini-batch weighted- mean is an unbiased estimate of the wo-weighted-mest i*.
E [Uer (i)) = CatwCa)( Big [+ (i))
(44)
w (i)) E Unit [45 [i)]
(45)
E Unit 1u (5)
(46)
Ay Unif ( (i)
(47)
11=1
n Ef Esg Unit (w(lWas)))))
(48)
n. B Unif
w (i
(49)
The expression in Equation (49) is invariant with respect to order of the indices ,,,..., iM: the indices in expectation in Equation (49) be exchanged, and thus has
Vk E [1, (1)) = n Eg Unif (ba) 2z (i)
(50)
Averaging Equation (50) over the indices & E [L, yields the desired result
24

---

M
M
Edi (i)
Σ M
Eg
Unif (eiltee u (1)
(51)
M
k=1
k=1
1
E0M (i)
n E
Unif
w
(1)
(52)
M
M
MM
5= (()) ((6))
n Eg Uni
(53)
M
Σラ w(b)
M
Σ f=1 w(())
n Eg Unii
Σ (aa)))
(54)
.M"
j= w(ll))
M
1 .
by Unie Σ (+ (bb))
n
E,
(55)
1 M
Σ
E
Unir
(56)
M
1
Σ
BiitU
Ei
(57)
M
1=1
1
M ENNUnif
(1))
(58)
M
EL-tnir
(59)
w
(60)
(61)
0*
(62)
25

---

Proof of Item (ii)). Using the same ideas as for Item (ii), one has
(62)..... ...,M000-aaa
[ (Izz !)?
(63)
nle aa
(11) T
(i)2
(64)
" Unif
(tt) u (8)²
44 E [1. M]
(65)
M
E M
SM Unif
w (2)) ui (2)2
(66)
ΣN M
w(
2j=1 M
E iM Unif M
(67)
Σ j= wilt))
Σラ (0())) (u())(
S
Eg
in
Unif
((x))
by convexity of + x²
(68)
M
2j=1 (o(ks)
Σ ==1 w
1 M
Σ (8x)
n
Unif
(69)
Σ== wlbh)
M
n M
1
E
Σ
(1 (u(1))2
(70)
Eni (u
(71)
ENunni (u)
(72)
Hence
1100)paaa(((ar (02) MMM - (4+)² El-Unir [w0 (u
(ū²)²
(73)
which is exactly
Var pesss (00)....,,MM)ppp (Ke, [)] S Varg(1) Panna cond (xe, (1), (1,
(74)
C Additional experiments
We present below the results for the MNIST dataset. The conclusions atre the same for the CIFAR-10 and CelebA 64 x 64: regressing against a more deterministic velocity field does not hurt generalization. On the contrary, generalization (i.e., lower test FID) appears earlier during training.
For this experiment, we used the Unet with attention and timestep embedding from torche fm library, with the Adam optimizer and all the default parameters. We used pretrained classifier with 99% accuracy on MNIST (90% on FMNIST) as a dower-dimensional embedding size 128 to compute the FID between the test and the generated set.
Method
Ep.
Ep.
Ep.
Ep.
Ep. 5
Ep. 10
Ep. 15
Ep. 20
Ep. 25
CFM (EFM, M=1)
378.00
181.25
67.88
29.44
15.30
4.20
3.08
2.51
2.28
EFM, M=128
370.64
168.58
60.52
25.52
13.44
3.79
2.70
2.35
2.10
EFM, M=256
370.94
169.71
61.88
25.73
13.48
3.73
2.76
2.33
2.08
EFM, M=1024
369.72
168.43
60.28
24.24
12.26
3.30
2.67
2.17
1.84
Table I: FID FMNIST. FID scores across training epochs for conditional flow matching and empirical flow matching for multiple values of the number samples used to estimate the closed-form i".
26

---

Method
FID Ep.
FID Ep. 10
FID Ep. 50
FID Ep. 100
FID Ep. 200
CFM (EFM, M=1)
253.56
48.67
25.36
21.35
19.67
EFM, M-128
206.27
44.08
23.39
19.63
17.72
EFM, M-256
202.62
45.06
22.16
20.08
17.74
BFM, M=512
194.66
44.19
22.10
18.93
16.85
Table 2: FID FMNIST. FID scores acToss training epochs for conditional flow matching and empirical flow matching for multiple values of the number samples M used to estimate the closed-form O*.
D Experiments details
For all the experiment we used all the same learning hyperparameters, the default ones form Tong et al. (2024). The hyperparameter values are summarized in Table 3. The details specific to each figure are described in Appendices D.2 to D.5
# Channels
Batch Size
Learning Rate
EMA Decay
Gradient Clipping
128
128
0.0002
0.9999
Table 3: Learning hyperparameters for all the CIFAR-IO and CelebA 64 experiments.
D.1
Compute time
Given that regressing against an estimate of the closed-form, EFM, seems to improve on CFM, one may wonder what the additional cost induced by EFN. To alleviate the non-linearity of GPU computing (parallelism may cause some discontinuities in terms of costs), we ran an exhaustive set of timing experiments, varying the batch size and the EFM sample size. To summarize the measurements (numbers are given for an NVIDIA L4 GPU, on CIFAR-10), 10), denoting b the batch size and e the EFM sample size, the cost follows (4.3ms + 0.9ps). It can be also be seen as adding 2% for every 100 EFM samples. Or, for instance with a batch size of 256, 1.1 second will be due to the 256-sample forward/hackwarr, while the additional cost for EFM-1000 will be 230ms (around 17% of the cost) and for EFM-128 under 30ms (under 3%).
D.2 Figures la and 1c
For Figure la deep learning is involved: the datasets 2-moons and CIFAR-10 are loaded. Then, 256 points from po Pdata are drawn, and one computes the mean the cosine similarities between @*1(1 t)xro tzı,t) t) and woond(( t)zo tæ æ1,t) I1 zo. for each value of t e {0, 1/100, 2/100,....99/100)
No deep learning either is involved in Figure Ic: the Imagenette dataset is loaded and spatially subsampled resolution dim 8, dim 16, .... dim 256, i.e., with d
82 d 3 162 .... d 3 256 Then, as for Figure 1a, batches of 256 points from po and Pdata are drawn, and one computes the percentage of cosine similarities between û' ((1 t)ar + Lx,, 1) and rond ((1 t) 1 z x1,t) t) 21 Zo, that are larger than 0.9, for multiple time values t.
D.3 Figure 2
In Figure 2, networks are trained with a vanilla conditional flow matching, with the standard 34 million parameters U-Net for diffusion by Nichol and Dhariwal (2021), with default settings from the torch fm codebase + (Tong et al., 2024). Training uses the CFM loss. For this specific experiment, we removed the usual random flip transform, for üt to be simpler and easier to estimate by us. For each "data" subsampling of the dataset, we trained the model for 10a iterations, with a batch size of 128, i.e., we trained the models for 128 epochs.
'nttps: http://hhh....cccco/oo////aaaaaa
27

---

D.4 Figure3 3
In Figure 3, for each dataset (CIFAR-10 10 and CelebA x 64), one network trained using vanilla conditional flow matching with the default parameters of Tong et al. (2024) (the most important ones are recalled in Table 3). Then images are generated first following the closed-form formula of the optimal velocity field i' from 0 to And then following the velocity field learned with usual conditional flow matching ug from T to 1.
D.5
Figure 4
For experiments involving training on CIFAR-10 (Figures 2 and 3), we rely on the standard 34 million parameters U-Net for diffusion by Nichol and Dhariwal (2021), with default settings from the torchfn Em codebase (Tong et al., 2024). For each algorithm, the networks are trained for 500k iterations with batch size 128, i.e., 1280 epochs.
For CelebA 64 x 64 (Figure 3), we rely on the training script of pnpflow library² (Martin et al., 2025), which uses U-Net from Huang et al. (2021); Ho al. (2020).
'nttps://giite wttp://wwww..ceeme/e/e///aaeaeaa
28


Reviews:
[{"review_id": "Q7xcqFLq5o", "reviewer": "Reviewer_MQbh", "summary": "Flow-matching models train a neural network to predict a time-dependent velocity field that carries data points to noise and, in practice, for some unknown reason, they are able to easily generate novel samples. This paper asks why that happens and draws upon two common explanation: first, that the injected noise in the training objective (target stochasticity) forces the model to see many noisy variants and thereby generalize; second, that the neural network cannot exactly fit the closed-form (or memorization) solution for the optimal velocity field, so it learns a smoother field that naturally interpolates between data points. \nThe authors use the fact that, under perfect fitting, the optimal velocity field is available in closed form and the learned distribution will match the empirical data distribution. To do so, the authors train flow-matching networks and measure how closely the learned field matches the closed-form one at different timeframes. The authors also provide a new training loss that effectively tries to remove the target stochasticity in the flow matching loss and see that no difference occurs in the generalization capabilities of the model. Through these controlled experiments, the authors claim that the generalization does not occur because of target stochasticity.\n\nOn the other hand, the authors have an interesting experiment showing that with different dataset sizes, as one increase the size of the dataset a flow-matching model is trained on, while the difference between the closed form velocity field and the learned velocity field increases (the model becomes less and less of a perfect fit) the model obtains better sample quality and novelty. This in turn hints at the fact that the neural network, while not learning the optimal velocity field, learns something (possibly smoother) that generalizes and interpolates better. \nAlthough the paper does not fully explain why the neural parameterization prefers those smoother solutions, its experiments convincingly dismiss the target stochasticity-based hypothesis and underscore the importance of inductive bias in flow-matching models. The work is well written, the experiments are carefully designed, and the insights are strong enough, in my view, to justify acceptance.", "questions": "1. In L38, what is meant by the “deterministic regime” of the flow matching objective? Do the authors mean the time steps closer to the data distribution where the generated sample does not change that much?\n2. Can you please add the ticks for the x-axis in Figure 3 (the bar plots where each bar indicates the number of samples)", "limitations": "Highlighted in the weakness section.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n1. Figures 1, 3, and 4 clearly illustrate the controlled experiments, with well-explained setups and convincing results that support the paper’s claims.\n2. The authors address the recently identified limitations of FID by using DinoV2 embeddings for evaluation, demonstrating attention to detail.\n3. The hybrid model that combines the closed-form solution with a learned neural network is a novel and interesting approach for probing the behaviour of flow-matching models (see Figure 4). The figure was also quite insightful as it showed that the generalization indeed happens early in the generation process, as switching the model to the closed-form version of the velocity field at a threshold time of as little as $\\tau=0.3$ removes the generalization capability (this is also consistent with Figure 3).\n\n## Weaknesses\n1. While the paper highlights that generalization may arise from the “mismatch” between the neural network and the closed-form velocity field, it does not provide insight into the underlying inductive biases that make this mismatch beneficial. It is already known that the closed-form solution yields a perfectly fitted flow-matching model that memorizes and regurgitates the training data, making it ineffective for generalization. What would be more insightful is an exploration of which types of mismatch are beneficial for generalization, rather than simply observing that a mismatch exists.\n2. Section 4 changes the training loss to effectively remove the stochasticity in the training objective. While I do agree that the empirical flow matching loss is “less stochastic,” it is not at all deterministic in any sense. It still relies on a Monte Carlo approximation and I do not believe this experiment alone fully obviates target stochasticity as an explanation for generalization (although I think the rest of the results complement this and the explanations are intuitive and convincing overall). Can you please add an experiment showing training scenarios where the batch hyper parameter in the EFM loss is also swept over? That would effectively control how “stochastic” the Monte-Carlo approximation, and as a result, the loss is.\n3. I would be interested in seeing similar experiments not just for CIFAR-10 and CelebA, but also for MNIST, FMNIST, and TinyImagenet to ensure that the conclusions are general enough.", "was_revised": true, "final_justification": "All of the explanations provided seem reasonable and adequately address my questions. After considering these along with the discussions in the other threads, I have raised my score from borderline accept to accept.", "rebuttal": "We would like to thank Reviewer MQbh for the very positive feedback and for highlighting the clarity and care given the experiments. Below, we answer the specific questions.\n\n> 1. While the paper highlights that generalization may arise from the “mismatch” between the neural network and the closed-form velocity field, it does not provide insight into the underlying inductive biases that make this mismatch beneficial. It is already known that the closed-form solution yields a perfectly fitted flow-matching model that memorizes and regurgitates the training data, making it ineffective for generalization. What would be more insightful is an exploration of which types of mismatch are beneficial for generalization, rather than simply observing that a mismatch exists.\n\nThere are two important ideas here. As the reviewer mentions, a closed-form solution exists that only generates training data. Thus, models that generalize fail to “match” this solution. \n- Our **first contribution is to discard “target stochasticity” suggested by Vastola (2025) as a cause for this mismatch**: we show that this phenomenon is limited in realistic high-dimensional settings (Section 3.1), and that models still generalize well when target stochasticity is removed from the problem (Section 4).\n- As a **second contribution, we investigate \"which part\" of the mismatch is a key driver of generalization**. Section 3.2 highlights \"mismatches\" at two specific times $t$: $t \\approx 0.1$ and $t \\approx 1$. Section 3.3 shows that the mismatch at time $t \\approx 0.1$ is especially important for generalization: we conclude that failure to learn the optimal velocity at early time steps is the main driver of generalization. \n\n\n> 2. Section 4 changes the training loss to effectively remove the stochasticity in the training objective. While I do agree that the empirical flow matching loss is “less stochastic,” it is not at all deterministic in any sense. It still relies on a Monte Carlo approximation and I do not believe this experiment alone fully obviates target stochasticity as an explanation for generalization (although I think the rest of the results complement this and the explanations are intuitive and convincing overall). Can you please add an experiment showing training scenarios where the batch hyper parameter in the EFM loss is also swept over? \n\nWe did not display the experiment with larger batches for the target, because our **results showed that larger batches $M > 1000$ and beyond yield almost the same generalization performances**, and the curves  $M > 1000$ almost superpose. We have added the corresponding parameter sweep in the revised manuscript.\n\n> 3. I would be interested in seeing similar experiments not just for CIFAR-10 and CelebA, but also for MNIST, FMNIST, and TinyImagenet to ensure that the conclusions are general enough.\n\nWe present below the results for the MNIST dataset. The conclusion is the same as for the other datasets: **regressing against a more deterministic velocity field does not hurt generalization**. On the contrary, generalization (i.e., lower test FID) appears earlier during training.\n\nFor this experiment, we used the Unet with attention and timestep embedding from torchcfm library, with the Adam optimizer and all the default parameters. We used a pretrained classifier with 99% accuracy on MNIST as a lower-dimensional embedding of size 128 to compute the FID between the test set and the generated set. \n\n| Method                           | FID Epoch 1 | FID Epoch 2 | FID Epoch 3 | FID Epoch 4 | FID Epoch 5 | FID Epoch 10 | FID Epoch 15 | FID Epoch 20 | FID Epoch 25 |\n|:-------------------------------- | -----------:| -----------:| -----------:| -----------:| -----------:| ------------:| ------------:| ------------:| ------------:|\n| CFM (EFM, M=1)                   |     378.00 |     181.25 |      67.88 |     29.44 |     15.30 |      4.20 |      3.08 |        2.51 |      2.28 |\n| EFM, M=128                       |      370.64 |     168.58 |     60.52 |     25.52 |     13.44 |      3.79 |      2.70 |      2.35 |      2.10 |\n| EFM, M=256                       |     370.94 |     169.71 |     61.88 |     25.73 |     13.48 |      3.73 |      2.76 |      2.33 |      2.08 |\n| EFM, M=1024 |     369.72 |     168.43 |     60.28 |     24.24 |     12.26 |      3.30 |      2.67 |      2.17 |      1.84 |\n\nWe will add the results for FMNIST and tiny-imagenet by the end of the author/reviewer discussion period.\n\n\n### Questions\n- *In L38, what is meant by the “deterministic regime” of the flow matching objective?*: We call the \"deterministic regime\" the times for which the average velocity field $\\hat u$ only correlates with a single of the $n$ conditional fields $ u^\\mathrm{cond}$, corresponding to the plot in Fig 1a, right.\n- Adding x-axis in Figure 3 : this has been done"}, {"review_id": "sPZaug96Fz", "reviewer": "Reviewer_Byfq", "summary": "This paper targets explaining the source of generalization in flow matching related methods. \nThe authors argue that it is the noisy loss, rather than the noisy data, that contribute to the generalization ability in flow matching.\nThe authors conducted interesting experiments to investigate the behavior of the target velocity as a function of time, and conclude that the introducing randomness in the training loss would lead to better generalization.\nInspired by this idea, the authors proposed a novel flow matching loss called empirical flow matching (EFM) and performed experiments to verify the generalization ability of EFM.", "questions": "Please see the \"strength and weakness\" part.", "limitations": "The authors should discuss their computational cost with empirical evidences.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- Towards understanding the generalization ability of flow matching, this paper provides a very interesting perspective on the learned velocity as a weighted sum of the conditional velocities and analyze the behavior of the weights.\n- I find the empirical validations in Sec 3 make sense for explaining the behavior of the velocity and the resulting generalization ability,\n- The novel EFM loss function is also deemed as a major contribution of this paper.\n\n## Weaknesses\n- This paper does not give a clear mathematical definition for the \"generalization\" and build their frameworks purely on (somewhat vague) empirical results. For example, can the generalization ability be defined as the velocity matching error on the training set? So that the relationship between the generalization ability and the training set size / piecewise threshold can be discussed more mathematically.\n-  I agree with the authors that the $M$ additional samples can be efficiently tackled by vectorization and the same backpropagation depth. However, empirical evidence on computational time and memory should be provided, as EFM explicitly introduces additional costs.\n- The experiments are not sufficient to prove the superiority of EFM, as different methods almost converge as the training proceeds. Moreover, to the best of my knowledge, quantititive results for flow matching methods can have large randomness.\n- In my opinion, the generalization ability generally leads to more evidently better test results than training results. However, in Figure 5, the training results and test results are almost the same. So can the authors explain this phenomenon?", "was_revised": true, "final_justification": "The authors fully addressed my concerns, so I'd like to raise my score to 5.", "rebuttal": "We would like to thank Reviewer Byfq for the very positive feedback and highlighting the new perspectives on generalization provided by the paper. Below, we answer the specific concerns.\n\n> This paper does not give a clear mathematical definition for the \"generalization\" and build their frameworks purely on (somewhat vague) empirical results. For example, can the generalization ability be defined as the velocity matching error on the training set?\n\nMathematically defining and measuring the generalization of generative models is a challenging and still largely open problem. As per [1], there are three criteria: “quantifying to what extent images resemble those from the training set (fidelity), how well the generated samples span the full training distribution (diversity), and whether they are truly novel or are simply reproductions of training samples (memorization)”. \n\nStandard evaluation metrics like FID have documented flaws [1,2]. For ourselves, we aimed to go beyond standard practice by:\n- Using **more expressive and discriminative embeddings** (DINOv2) instead of the commonly used Inception features\n- Measuring (approximate) Wasserstein distance between the representations of the **generated set and the test set**, rather than the training set, as it is usual done in generative modeling benchmarks\n- Displaying a **memorization metric** that would detect a pure copy of the training set\n\nWe have extended the discussion in the Metrics paragraph L244 to clarify this discussion.\n\n> I agree with the authors that the additional samples can be efficiently tackled by vectorization and the same backpropagation depth. However, empirical evidence on computational time and memory should be provided, as EFM explicitly introduces additional costs.\n\n- We emphasize that our main contribution is to challenge target stochasticity as the main driver of generalization: the main point of Section 4 is that the more deterministic EFM works at least as well as the stochastic CFM.\n-  However, given that EFM turns out to improve on CFM, its cost indeed becomes very interesting. To alleviate the non-linearity of GPU computing (parallelism may cause some discontinuities in terms of costs), we ran an exhaustive set of timing experiments, varying the batch size and the EFM sample size. To summarize the measurements (numbers are given for an NVIDIA L4 GPU, on CIFAR-10), denoting $b$ the batch size and $e$ the EFM sample size, the cost follows approximately $b \\times ( 4.1\\mathrm{ms} + e \\times 0.04 \\mathrm{μs})$. It can be also be seen as adding ~1% for every 1000 EFM samples. Or, for instance with a batch size of 256, 1.05 sec will be due to the 256-sample forward/backward, while the additional cost for EFM-1000 will be 10ms (around 1%) and for EFM-128, under 1.3 ms.\n\n> The experiments are not sufficient to prove the superiority of EFM, as different methods almost converge as the training proceeds. Moreover, to the best of my knowledge, quantititive results for flow matching methods can have large randomness.\n\nWe agree with the reviewer, and we emphasize that **our claim is not that the EFM should become the new standard paradigm for training**. We only mean to show that, when target stochasticity is removed from the problem (as in the EFM objective), models still generalize well, reaching as low FID as other training methods. This rules out target stochasticity as the driver for generalization. We added MNIST and will add FMNIST and tiny_imagenet by the end of the reviewers-authors discussion period, to strengthen this statement (see answer to reviewer MQbh).\n\n> In my opinion, the generalization ability generally leads to more evidently better test results than training results. However, in Figure 5, the training results and test results are almost the same. So can the authors explain this phenomenon?\n\nIn all our experiments, **we indeed observe a strong correlation between FID scores on the training and test sets**. This suggests that the models generalize well and that there is no significant overfitting in terms of FID. We believe this strong correlation between train and test FID may also reflect a limitation of the FID metric itself: FID does not penalize overfitting and does not detect subtle generalization gaps. This has been discussed in prior work (e.g., [2]).\n\nWhile the close alignment between train and test FID may initially seem surprising, it is consistent with prior observations in generative model evaluation and reinforces the practice of using train FID as a proxy for test performance, though, as always, with some caution.\n\n\n**[1]** Stein, G., Cresswell, J., Hosseinzadeh, R., Sui, Y., Ross, B., Villecroze, V., Liu, Z., Caterini, A.L., Taylor, E. and Loaiza-Ganem, G., 2023. Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models. Neurips.\n\n**[2]** Jiralerspong, M., Bose, J., Gemp, I., Qin, C., Bachrach, Y. and Gidel, G., 2023. Feature likelihood divergence: evaluating the generalization of generative models using samples. Neurips"}, {"review_id": "PeUKfJaSYq", "reviewer": "Reviewer_ik8M", "summary": "The paper analyses the reasons for generalization of flow matching generative models. Its main focus is ruling out a recent explanation identifying the stochasticity of the target as the root for generalization of flow matching. The authors make use of the closed form optimal solution of the velocity field to eliminate the stochasticity and show that although the target is deterministic, the model still generalizes (and even surprisingly performs better).", "questions": "- Can the authors extend the discussion on how the error in approximating the velocity relates to the dynamical regimes found in [\\[2402.18491\\] Dynamical Regimes of Diffusion Models](https://arxiv.org/abs/2402.18491) ?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- Clarity: the paper is well written, the problem and the experimental settings are clear.   \n- Originality: the paper takes a courageous approach challenging another work’s conclusions and it does so respectfully and backed up with thorough experimental results.   \n- Removing loss stochasticity could simplify implementation and improve Flow Matching model training stability/efficiency.\n\n**Weaknesses**\n\n- The analysis is done only over image modality and with U-Net architecture. Would be interesting to see if the same behavior happens for other modalities/architectures and how this affects the error in approximating the velocity.", "was_revised": true, "final_justification": "I keep my originally given high score for this paper and recommend acceptance.", "rebuttal": "We would like to thank Reviewer ik8M for the very positive feedback and for highlighting the clarity and originality of the manuscript. Below, we answer the specific questions.\n\n> The analysis is done only over image modality and with U-Net architecture. Would be interesting to see if the same behavior happens for other modalities/architectures and how this affects the error in approximating the velocity.\n\nWe agree with the reviewer that our paper specifically targets image generation, as this is the main application for Flow Matching. In all the experiments, we used the reference Unet with attention and timestep embedding from the reference torchcfm library. \nWe believe that we would obtain the same results with other architectures : for diffusion, [1] have shown that multiple architectures (DiT, U-ViT DDPM++, NCSN) end up learning the same velocity field (see their Fig. 1 and Fig 9)\n\n[1] Towards a Mechanistic Explanation of Diffusion Model Generalization, ICML 2025, Niedoba, Matthew and Zwartsenberg, Berend and Murphy, Kevin and Wood, Frank\n\n\n> Can the authors extend the discussion on how the error in approximating the velocity relates to the dynamical regimes found in \"Dynamical Regimes of Diffusion Models\" ?\n\n- The analysis in \"Dynamical Regimes of Diffusion Models\" [Biroli et al., 2024] focuses on the behavior of the **closed-form score function** in diffusion models, **without considering how neural networks actually learn or approximate this function**. Their main contribution is to identify three dynamical regimes—noisy, speciation, and collapse—that emerge when sampling is performed using the exact, analytical score in a Gaussian mixture setting. These regimes are derived from connections to classical phase transitions in statistical physics.\n- In contrast, our work investigates **how deep learning models approximate this ideal score function**, and more importantly, **how this failure to approximate the \"optimal\" score impact generalization**. We empirically observe that model errors are concentrated around two critical timesteps (Figure 3), which may correspond to key dynamical transitions.\n- Regarding the study of the non-stochasticity of the velocity field, we added experiments showing how the critical time for non-stochasticity (defined as a cosine similarity between $u^\\text{cond}$ and $\\hat u$ greater than 0.9) varies with both the number of training samples $n$ and  the data dimension $d$. We observe a dependence in $\\frac{\\log n}{d}$, which is the same as the one obtained by Biroli for their 'collapse time'  (defined as the time from which the diffusion trajectories are attracted by a single training point). \n\nDrawing connections between the empirical failure modes we observe and the theoretical regimes identified by Biroli et al. is a promising direction for future work. Such a connection would require extending their theoretical framework to account for learning dynamics and approximation errors introduced by neural architectures."}, {"review_id": "wAXT66mpAB", "reviewer": "Reviewer_xahj", "summary": "This manuscript investigates whether the stochasticity introduced by the conditional flow matching loss contributes to model generalization. While prior work suggests that conditioning on vector fields induces beneficial randomness, the authors demonstrate that, in high-dimensional settings, this stochastic effect occurs only within a narrow time window, casting doubt on its generalization benefit. To address this, they propose a novel training algorithm that interpolates between conditional and unconditional flow matching losses, empirically validating that even limited stochasticity can enhance generalization performance.", "questions": "1. In line 89, the notation $z\\sim p_0\\times\\hat{p_{data}}$ is ambiguous. Do you intend that z is sampled from $\\hat{p}_{data}$ while an independent noise term is drawn from $p_0$?\n2. Figure 5 is difficult to interpret: the gray curve is nearly invisible, and the reported gains from EFM appear marginal. I recommend reformatting the results—perhaps via a table or clearer color scheme—to improve readability.\n3. Regarding the loss in Equation (7), can it be viewed as an interpolation between the conditional and unconditional flow matching losses? In particular, setting M=1 recovers the conditional loss, while M=the number of data seems to correspond to the unconditional loss.", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n\n- The paper offers compelling empirical insights into how conditional vector fields differ from their unconditional counterparts, deepening our understanding of flow matching training dynamics.\n- The proposed interpolation framework between loss functions is novel and may inspire future theoretical developments.\n\n**Weaknesses**\n\n- The manuscript’s narrative lacks coherence: the title and early claims argue against stochasticity as a driver of generalization, yet Section 4 appears to contradict this by showing that reducing stochasticity can improve performance. A unified, logically consistent storyline is needed.", "was_revised": true, "final_justification": "I had a misinterpretation on the purpose of section 4 during my review. The authors pointed this out and I agree with what they said. They have hence resolved my biggest concern. I like the results of this paper as it provides a concise answer to an issue raised not long ago and this paper may be helpful for the community in understanding the generalization of flow matching.", "rebuttal": "We would to thank Reviewer xahj for the very positive feeback, highlighting the compelling empirical analysis. Below, we provide answer to the specific questions.\n\n> The title and early claims argue against stochasticity as a driver of generalization, yet Section 4 appears to contradict this by showing that reducing stochasticity can improve performance.\n\n\nWe are confused by this sentence; in our opinion **both the title and Section 4 consistently support the same message**:\n- **The title argues against target stochasticity** as a key driver of generalization\n- **Section 4 supports that target stochasticity does not bring generalization improvement** compared to its more deterministic counterpart. Our findings tend to support that regressing against a deterministic target (EFM, Fig 5) yields generalization earlier during training.  In particular, Section 4 shows that the more deterministic the target (i.e., the larger the number of samples $M$ to estimate the closed-form formula), the earlier the generalization occurs.\n\nWe would be grateful if the reviewer could elaborate on which part of the narrative appeared contradictory, so we can address it more explicitly in the revised manuscript.\n\n\n> 1. In line 89, the notation “$z \\sim p_0 \\times \\hat p_\\mathrm{data}$” is ambiguous. Do you intend that $z$ is sampled from $\\hat p_\\mathrm{data}$ while an independent noise term is drawn from $p_0$ ?\n\n\n What we meant here is that instead of choosing as conditioning variable $z = x^{(i)} \\sim \\hat p_\\mathrm{data}$, an alternate choice (used eg by Albergo and Vanden Eijnden 2023, or Liu et al 2023) is to use $z = (x_0, x^{(i)})$ where $x_0$ is noise. This sentence has been removed for clarity.\n\n\n> 2. Figure 5 is difficult to interpret: the gray curve is nearly invisible, and the reported gains from EFM appear marginal. I recommend reformatting the results—perhaps via a table or clearer color scheme—to improve readability.\n\t\nReadability has been improved using larger figures. However, we would like to underscore that the goal of this figure is not to show that \"Empirical flow matching (EFM) works better\" than conditional flow matching. The core idea is to show that the estimator based on the stochastic target (vanilla conditional flow matching) and the estimator with a more deterministic target (empirical flow matching) yield very similar generalization performance. Hence, **target stochasticity is not the driver of generalization.**\n\nIn addition, as the reviewer observed the curves for $M=256$ and $M=1000$ are almost identical, which shows that above this threshold, using more points in EFM does not affect the performance. We have made this clearer in the discussion of the experiments.\n\n> 3. Regarding the loss in Equation (7), can it be viewed as an interpolation between the conditional and unconditional flow matching losses? In particular, setting M=1 recovers the conditional loss, while M=the number of data seems to correspond to the unconditional loss.\n\nYes! This is exactly the idea! \n- $M=1$ recovers the vanilla conditional flow matching loss, with a stochastic target. \n- $M = \\text{number of samples}$ recovers fully the flow matching loss, with a deterministic target, that would be very costly to compute in practice\n- $1 < M < \\text{number of samples}$ is a stochastic, but more deterministic target that is cheaper to compute in practice ($\\approx 1$% overhead for $M=1000$). Please also refer to the discussion with Reviewer Byfq for additional details on the timings."}]

Author rebuttal:

Meta-review: (a) This paper investigates why flow-matching generative models generalize rather than memorizing training data. Two popular hypotheses are tested: 1) target stochasticity in the training loss drives generalization and 2) neural approximation mismatch with the closed-form velocity field induces smoother fields that interpolate between samples. Using the closed-form velocity field, the authors show that perfect fitting leads to memorization, whereas learned approximations deviate and yield better generalization. They introduce Empirical Flow Matching (EFM), a loss formulation that reduces stochasticity in training, and find that generalization still persists, thus refuting the stochasticity hypothesis. These claims are also verified experimentally.

(b) The method introduces a novel and insightful use of controlled loss variants to probe the role of stochasticity. The hypotheses are also validated by carefully designed and clearly presented experiments with convincing visualizations. The paper is well written. The proposed EFM loss provides a practically simpler alternative with stable training.

(c) The reviewers notice that the paper is mostly empirical in nature, as it does not provide a theoretical insight into why the mismatch between neural networks and the closed-form solution leads to generalization. The experiments could include a broader range of modalities, datasets and architectures. Some reviewers noted a lack of consistency in the narrative between rejecting and partially retaining the role of stochasticity.

(d) I recommend acceptance. All reviewers converged on positive assessments after rebuttal, with scores uniformly raised to 5. The paper provides meaningful contributions: it challenges a widely discussed explanation of generalization in flow-matching, proposes a simple alternative training loss, and supplies careful empirical evidence. While theoretical depth and broader validation are limited, the work clearly advances understanding in this area and is of interest to the generative modeling community. 

(e) Across reviews, concerns centered on 1) the exact role of stochasticity in EFM, 2) missing computational cost evaluation, 3) limited dataset and modality scope, and 4) some presentation/narrative clarity issues. The authors clarified most of the important issues and the reviewers found these responses satisfactory. In fact, they raised their scores after rebuttal, with several stating that their primary concerns were resolved. The outstanding limitations (lack of broader experiments and deeper theoretical analysis) were acknowledged but not deemed critical for acceptance.
Decision: Accept (oral)