Title: On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity
Venue: NeurIPS 2025
Accepted: yes
Keywords: flow matching

generalization

memorization
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\n1. 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.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-30"}

comments: 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).

STRENGTHS AND WEAKNESSES:
**Strengths**

- Clarity: the paper is well written, the problem and the experimental settings are clear.   
- Originality: the paper takes a courageous approach challenging another work’s conclusions and it does so respectfully and backed up with thorough experimental results.   
- Removing loss stochasticity could simplify implementation and improve Flow Matching model training stability/efficiency.

**Weaknesses**

- 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.

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
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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?\n\nQUESTIONS:\nPlease see the \"strength and weakness\" part.\n\nLIMITATIONS:\nThe authors should discuss their computational cost with empirical evidences.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nFlow-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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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.\n\nQUESTIONS:\n1. 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)\n\nLIMITATIONS:\nHighlighted in the weakness section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-07"}