Continuous-time generative models, such as Flow Matching (FM), construct
probability paths to transport between one distribution and another through the
simulation-free learning of the neural ordinary differential equations (ODEs).
During inference, however, the learned model often requires multiple neural
network evaluations to accurately integrate the flow, resulting in a slow
sampling speed. We attribute the reason to the inherent (joint) heterogeneity
of source and/or target distributions, namely the singularity problem, which
poses challenges for training the neural ODEs effectively. To address this
issue, we propose a more general framework, termed Switched FM (SFM), that
eliminates singularities via switching ODEs, as opposed to using a uniform ODE
in FM. Importantly, we theoretically show that FM cannot transport between two
simple distributions due to the existence and uniqueness of initial value
problems of ODEs, while these limitations can be well tackled by SFM. From an
orthogonal perspective, our framework can seamlessly integrate with the
existing advanced techniques, such as minibatch optimal transport, to further
enhance the straightness of the flow, yielding a more efficient sampling
process with reduced costs. We demonstrate the effectiveness of the newly
proposed SFM through several numerical examples.

通过使用切换的普通微分方程 (ODEs) 来消除奇点问题，我们提出了一个更通用的框架，Switched FM (SFM)，以解决连续时间生成模型中的采样速度缓慢的问题，并演示了该框架的有效性。

切换流匹配：通过切换 ODE 来消除奇异性

Switched Flow Matching: Eliminating Singularities via Switching ODEs

Straightening the probability flow of the continuous-time generative models,
such as diffusion models or flow-based models, is the key to fast sampling
through the numerical solvers, existing methods learn a linear path by directly
generating the probability path the joint distribution between the noise and
data distribution. One key reason for the slow sampling speed of the ODE-based
solvers that simulate these generative models is the global truncation error of
the ODE solver, caused by the high curvature of the ODE trajectory, which
explodes the truncation error of the numerical solvers in the low-NFE regime.
To address this challenge, We propose a novel method called SeqRF, a learning
technique that straightens the probability flow to reduce the global truncation
error and hence enable acceleration of sampling and improve the synthesis
quality. In both theoretical and empirical studies, we first observe the
straightening property of our SeqRF. Through empirical evaluations via SeqRF
over flow-based generative models, We achieve surpassing results on CIFAR-10,
CelebA-$64 \times 64$, and LSUN-Church datasets.

SeqRF 是一种学习技术，通过改善概率流的直线性来减少全局截断误差，加速采样并提高合成质量。在理论和实证研究中，我们观察到了 SeqRF 的直线性属性，并通过 SeqRF 在基于流模型的生成模型上进行的实证评估，在 CIFAR-10、CelebA-$64 	imes 64$ 和 LSUN-Church 数据集上取得了优异的结果。

生成模型的顺序流匹配

Sequential Flow Matching for Generative Modeling

Simulation-free methods for training continuous-time generative models
construct probability paths that go between noise distributions and individual
data samples. Recent works, such as Flow Matching, derived paths that are
optimal for each data sample. However, these algorithms rely on independent
data and noise samples, and do not exploit underlying structure in the data
distribution for constructing probability paths. We propose Multisample Flow
Matching, a more general framework that uses non-trivial couplings between data
and noise samples while satisfying the correct marginal constraints. At very
small overhead costs, this generalization allows us to (i) reduce gradient
variance during training, (ii) obtain straighter flows for the learned vector
field, which allows us to generate high-quality samples using fewer function
evaluations, and (iii) obtain transport maps with lower cost in high
dimensions, which has applications beyond generative modeling. Importantly, we
do so in a completely simulation-free manner with a simple minimization
objective. We show that our proposed methods improve sample consistency on
downsampled ImageNet data sets, and lead to better low-cost sample generation.

通过非简单合并数据和噪声样本的 Multisample Flow Matching 算法，我们可以在完全不需要模拟的情况下，通过一个简单的最小化目标函数构造概率路径，实现连续时间生成模型训练参数、生成高质量样本和高维传输映射等目标的综合优化。