Flow Matching（流匹配）Flow Matching

机器人策略要输出的往往不是“一个动作”，而是“动作的分布”。同一个场景下，示教者有时从左边绕开障碍，有时从右边；同一个杯子可以从好几个方向抓。19.1 节已经看到，用均方误差回归动作会学到两个模式的平均，正好撞上障碍物。所以我们需要一个能表达多峰分布、又能快速采样的生成模型。

19.6 节的扩散模型能做到前一半：它把数据一步步加噪成高斯，再学习如何一步步去噪，样本质量很好，Diffusion Policy（22.2 节）就是这样生成动作的。但机器人有一个很硬的约束：时间。一个以 50 Hz 运行的控制器，每 20 ms 就要给出下一个动作；如果采样一次要几十到上百次网络前向，就很难赶上。19.3 节的连续 flow 是另一条路：沿一个向量场积分 ODE 就能采样，可是按最大似然训练它，每一步训练都要把 ODE 积分一遍。

Flow matching（Lipman 等，2023；同期的 rectified flow 与 stochastic interpolants 是同一类想法）把两者的优点拼在一起：它学的也是一个把噪声“流”成数据的速度场 \(\mathbf v_\theta(\mathbf x,t)\)，但训练时不积分，只是对一个随手就能写出来的目标做回归；而且这个目标来自直线路径，所以通常几步积分就够了。π0 这类视觉-语言-动作模型的动作头用的就是它：每次推理只积分 10 步，就生成一整段 50 步的动作块。

What a robot policy must output is often not "an action" but "a distribution over actions". In the same scene a demonstrator sometimes passes an obstacle on the left and sometimes on the right; the same cup can be grasped from several directions. Lesson 19.1 already showed that regressing actions with a mean squared error learns the average of the two modes, which drives straight into the obstacle. So we need a generative model that can represent multi-peaked distributions and can also sample fast.

The diffusion models of Lesson 19.6 do the first half: they add noise to the data step by step until it is Gaussian and learn to remove it step by step; sample quality is excellent, and Diffusion Policy (Lesson 22.2) generates actions exactly this way. But robots face a hard constraint: time. A controller running at 50 Hz must produce its next action every 20 ms, and a sampler that needs tens to hundreds of network passes struggles to keep up. The continuous flows of Lesson 19.3 are another road: sampling means integrating an ODE along a vector field, but training them by maximum likelihood integrates the ODE at every training step.

Flow matching (Lipman et al., 2023; the concurrent rectified flow and stochastic interpolants are the same family of ideas) combines the strengths of both. It also learns a velocity field \(\mathbf v_\theta(\mathbf x,t)\) that "flows" noise into data, but training never integrates anything: it is regression onto a target you can write down directly. And because that target comes from straight paths, a few integration steps are usually enough. The action heads of vision-language-action models such as π0 use exactly this: each inference integrates only 10 steps and produces a whole chunk of 50 future actions.

3.6 节把向量场想成一条河：撒一把沙子，每粒沙子顺着水流漂。现在我们想设计一条河，让 \(t=0\) 时撒下的一团高斯沙子 \(p_0=\mathcal N(\mathbf 0,I)\)，到 \(t=1\) 时恰好堆成数据的形状 \(q\)。最笨也最直接的办法：随机配对，每粒噪声 \(\mathbf x_0\sim p_0\) 配一个数据点 \(\mathbf x_1\sim q\)，让它沿直线匀速走过去：

Lesson 3.6 pictured a vector field as a river: sprinkle sand and each grain drifts with the current. Now we want to design a river so that a Gaussian cloud of sand sprinkled at \(t=0\), \(p_0=\mathcal N(\mathbf 0,I)\), has piled up into the shape of the data \(q\) by \(t=1\). The bluntest, most direct recipe: pair at random, give each grain of noise \(\mathbf x_0\sim p_0\) a data point \(\mathbf x_1\sim q\), and let it walk there along a straight line at constant speed:

在每个时刻 \(t\)，所有沙粒的位置构成一个分布 \(\hl{pt}{p_t}\)，从 \(p_0\)（高斯）连续地变到 \(p_1=q\)（数据）。这一串分布叫概率路径。问题在于：这些直线会交叉。两条直线在同一个时刻穿过同一点，却要朝不同方向走，而一个向量场在一个点只能给出一个速度。解决办法就是本节的核心：在每个点 \((\mathbf x,t)\)，取所有经过这里的直线速度的平均，

At each time \(t\) the positions of all grains form a distribution \(\hl{pt}{p_t}\) that moves continuously from \(p_0\) (the Gaussian) to \(p_1=q\) (the data). This family of distributions is called a probability path. The trouble is that the lines cross. Two lines pass through the same point at the same time heading in different directions, yet a vector field can give only one velocity per point. The fix is the heart of this lesson: at every point \((\mathbf x,t)\), take the average of the velocities of all lines that pass through it,

第二个等号用了 \(\mathbf x_0=(\mathbf x-t\mathbf x_1)/(1-t)\)：知道了现在的位置和终点，起点就确定了。\(p(\mathbf x_1\mid\mathbf x_t=\mathbf x)\) 是“一粒此刻在 \(\mathbf x\) 的沙子，终点是哪里”的后验。一个不显然、却是本节最重要的事实是：沿这个平均速度场漂流，整团沙子在每个时刻的分布仍然恰好是 \(p_t\)（第 5 节证明）。单粒沙子的路线变了：它们不再走直线，而是走一条弯曲、互不交叉的曲线，但整体的“人群密度”与直线配对时一模一样。下图在一维里把两者画在一起。横轴是时间，纵轴是位置。

The second equality uses \(\mathbf x_0=(\mathbf x-t\mathbf x_1)/(1-t)\): once you know where a grain is now and where it ends, its start is fixed. \(p(\mathbf x_1\mid\mathbf x_t=\mathbf x)\) is the posterior "where does a grain that is at \(\mathbf x\) right now end up". The non-obvious fact, and the most important one of this lesson, is this: drifting along this averaged field moves the whole cloud so that its distribution at every time is still exactly \(p_t\) (proved in Section 5). Individual grains take different routes: no longer straight lines but curved paths that never cross, yet the overall "crowd density" is identical to that of the straight-line pairing. The figure puts both side by side in one dimension: time runs horizontally, position vertically.

把上面的配方原样写成代码，就是条件 flow matching（CFM）的训练循环。每一步：抽一批噪声 \(\mathbf x_0\sim\mathcal N(\mathbf 0,I)\)、一批数据 \(\mathbf x_1\sim q\) 和时间 \(t\sim\mathcal U[0,1]\)，算出 \(\mathbf x_t\)，让网络在那里预测速度，与直线速度 \(\mathbf x_1-\mathbf x_0\) 比平方误差：

Write that recipe down as code and you have the training loop of conditional flow matching (CFM). At each step, draw a batch of noise \(\mathbf x_0\sim\mathcal N(\mathbf 0,I)\), a batch of data \(\mathbf x_1\sim q\) and times \(t\sim\mathcal U[0,1]\), form \(\mathbf x_t\), let the network predict a velocity there, and compare it with the line's velocity \(\mathbf x_1-\mathbf x_0\) in squared error:

没有 ODE、没有雅可比行列式、没有噪声调度表：就是一个回归问题，梯度用 16.3 节的反向传播。下图的网络是一个 \([5,48,48,2]\) 的 GELU MLP，输入是 \((x,y,t,\sin\pi t,\cos\pi t)\)，共 2738 个参数；优化器是 Adam（学习率 0.003，2000 步后降到 0.001），每步 64 个样本。数据是二维的高斯混合，这样页面能算出精确的平均速度场 \(\mathbf u_t\)（第 5 节给出公式），于是我们可以直接量出网络离理想答案还有多远：

No ODE, no Jacobian determinant, no noise schedule: just a regression problem, with gradients from the backpropagation of Lesson 16.3. The network in the figure is a \([5,48,48,2]\) GELU MLP whose input is \((x,y,t,\sin\pi t,\cos\pi t)\), 2738 parameters in all; the optimizer is Adam (learning rate 0.003, lowered to 0.001 after 2000 steps) on batches of 64. The data are 2-D Gaussian mixtures, so the page can compute the exact averaged field \(\mathbf u_t\) (the formula is in Section 5) and measure directly how far the network still is from the ideal answer:

第一项是场误差 \(\mathcal E(\theta)\)，训练能把它压向零；第二项 \(\mathcal L^*\) 与 \(\theta\) 无关，是直线速度围绕平均值的散布：同一点上直线们彼此越不一致，它越大。所以 CFM 的损失永远不会降到零，就像 16.2 节验证损失有一个噪声地板一样。采样时从 500 个固定的高斯样本出发，用 Euler 法积分 \(\mathbf x\leftarrow\mathbf x+h\,\mathbf v_\theta(\mathbf x,t)\)，\(h=1/N\)（2.5、3.6 节）。

The first term is the field error \(\mathcal E(\theta)\), which training can push towards zero; the second, \(\mathcal L^*\), does not depend on \(\theta\): it is the scatter of the line velocities around their average, larger wherever the lines through a point disagree more. So the CFM loss never reaches zero, just as the validation loss in Lesson 16.2 has a noise floor. Sampling starts from 500 fixed Gaussian samples and integrates \(\mathbf x\leftarrow\mathbf x+h\,\mathbf v_\theta(\mathbf x,t)\) with Euler's method, \(h=1/N\) (Lessons 2.5 and 3.6).

训练好之后，生成一个样本就是解一个 ODE：\(\dot{\mathbf x}=\mathbf v_\theta(\mathbf x,t)\)，从 \(t=0\) 积到 \(t=1\)。每次网络前向（NFE，number of function evaluations）都要花时间，所以问题是：给定一个 NFE 预算，样本有多准？下图用精确的平均场（不是网络）把“学得好不好”和“积分得准不准”分开，只看后者。除了直线路径，也可以选一条余弦路径 \(\mathbf x_t=\sin(\tfrac{\pi t}{2})\mathbf x_1+\cos(\tfrac{\pi t}{2})\mathbf x_0\)：它的信号和噪声系数平方和为 1，与 19.6 节扩散模型常用的“方差保持”型路径属于同一类（第 5 节）；把时间反过来（\(t=1-\tau/T\)），它就是 19.6 节的余弦日程（略去那里的小偏移 \(s\)），\(\sin\tfrac{\pi t}{2}\approx\sqrt{\bar\alpha_\tau}\)。积分方法可选 Euler（每步 1 次 NFE，一阶）或中点法（每步 2 次 NFE，二阶，3.6 节 FIG 4 里的 Heun 也是二阶）。

Once trained, generating a sample means solving an ODE, \(\dot{\mathbf x}=\mathbf v_\theta(\mathbf x,t)\), from \(t=0\) to \(t=1\). Every network pass (NFE, number of function evaluations) costs time, so the question is: for a given NFE budget, how accurate is the sample? The figure uses the exact averaged field (not a network) to separate "how well was it learned" from "how accurately is it integrated", and looks only at the latter. Besides the straight path you can pick a cosine path, \(\mathbf x_t=\sin(\tfrac{\pi t}{2})\mathbf x_1+\cos(\tfrac{\pi t}{2})\mathbf x_0\): its signal and noise coefficients have squares summing to 1, the same family as the "variance-preserving" paths common in the diffusion models of Lesson 19.6 (Section 5); with time reversed (\(t=1-\tau/T\)) it is 19.6's cosine schedule (dropping its small offset \(s\)), \(\sin\tfrac{\pi t}{2}\approx\sqrt{\bar\alpha_\tau}\). The solver is Euler (1 NFE per step, first order) or the midpoint method (2 NFE per step, second order; the Heun method of Lesson 3.6's FIG 4 is second order too).

答案是余弦路径 + Euler（0.141，直线 + Euler 是 0.177，两种中点法都在 0.26 以上）。这值得停下来想一想。直线路径的条件路径是直的，但独立随机配对时，平均之后的边缘轨迹仍然是弯的，FIG 3 的青色曲线就是证据；在这个例子里，余弦路径配上等间距的时间步反而更好积分。真实的采样器还会把时间步排得不均匀，所以“哪条路径更省步数”没有放之四海皆准的答案。直线路径真正的优势在别处：如果配对本身是“不交叉”的，直线路径下的边缘轨迹就真的是直线，一步 Euler 就精确。下一张图就去造这样的配对。

The answer is cosine path + Euler (0.141; straight + Euler gives 0.177, and both midpoint variants are above 0.26). This deserves a pause. The conditional paths of the straight path are straight, but with independent random pairing the averaged marginal trajectories are still curved, as FIG 3's cyan curves show; in this example the cosine path with evenly spaced time steps is actually easier to integrate. Practical samplers also space their time steps unevenly, so "which path saves steps" has no universal answer. The real advantage of the straight path lies elsewhere: if the pairing itself is "non-crossing", the marginal trajectories of the straight path are straight lines, and a single Euler step is exact. The next figure builds such pairings.

边缘轨迹之所以弯，是因为训练时的直线互相交叉，平均之后拐了弯。交叉来自随机配对：一粒在左上的噪声，可能被配给右下的数据点。两种办法能让配对“有序”：

• 小批量最优传输（Tong 等，2024；Pooladian 等，2023）：每个批次里有 64 个 \(\mathbf x_0\) 和 64 个 \(\mathbf x_1\)，不再随机配，而是求一个让总平方距离 \(\sum_i\|\mathbf x_0^{(i)}-\mathbf x_1^{(\sigma(i))}\|^2\) 最小的一一配对 \(\sigma\)（指派问题，本页用匈牙利算法精确求解）。最优传输的配对线段互不交叉。
• Reflow（Liu 等，2023 的 rectified flow）：先训练一个普通的 flow，然后从噪声 \(\mathbf x_0\) 出发积分它得到 \(\mathbf x_1=\mathrm{ODE}(\mathbf x_0)\)，把这些 \((\mathbf x_0,\mathbf x_1)\) 当作新的配对再训练一次。ODE 的轨迹互不交叉，所以新的配对也不交叉，新的流更直。本页为了省时间，直接用精确平均场的 ODE（RK4，10 步）生成 2000 对，也就是 FIG 2 那个网络努力逼近的流。

The marginal trajectories are curved because the training lines cross each other and bend when averaged. The crossings come from random pairing: a grain of noise at the top left may be paired with a data point at the bottom right. Two ways to make the pairing "orderly":

• Minibatch optimal transport (Tong et al., 2024; Pooladian et al., 2023): a batch holds 64 samples \(\mathbf x_0\) and 64 samples \(\mathbf x_1\); instead of pairing them at random, find the one-to-one pairing \(\sigma\) that minimizes the total squared distance \(\sum_i\|\mathbf x_0^{(i)}-\mathbf x_1^{(\sigma(i))}\|^2\) (an assignment problem, solved exactly on this page with the Hungarian algorithm). The segments of an optimal-transport pairing never cross.
• Reflow (the rectified flow of Liu et al., 2023): first train an ordinary flow, then integrate it from noise \(\mathbf x_0\) to get \(\mathbf x_1=\mathrm{ODE}(\mathbf x_0)\), and train again on these \((\mathbf x_0,\mathbf x_1)\) as the new pairs. ODE trajectories never cross, so neither do the new pairs, and the new flow is straighter. To save time, this page generates the 2000 pairs directly from the ODE of the exact averaged field (RK4, 10 steps), which is the very flow FIG 2's network tries to approximate.

记 \(\mathbf z=(\mathbf x_0,\mathbf x_1)\) 为一对配对，按某个联合分布 \(\pi(\mathbf z)\) 抽取；独立配对时 \(\pi=p_0\otimes q\)。每一对给出一条条件路径：\(\mathbf x_t=\psi_t(\mathbf z)=(1-t)\mathbf x_0+t\mathbf x_1\)，速度 \(\hl{cond}{\mathbf u_t(\mathbf x\mid\mathbf z)=\mathbf x_1-\mathbf x_0}\)。对所有配对平均，就得到边缘的概率路径 \(\hl{pt}{p_t}\) 和边缘速度场：

Let \(\mathbf z=(\mathbf x_0,\mathbf x_1)\) be a pair drawn from some joint distribution \(\pi(\mathbf z)\); with independent pairing \(\pi=p_0\otimes q\). Each pair defines a conditional path \(\mathbf x_t=\psi_t(\mathbf z)=(1-t)\mathbf x_0+t\mathbf x_1\) with velocity \(\hl{cond}{\mathbf u_t(\mathbf x\mid\mathbf z)=\mathbf x_1-\mathbf x_0}\). Averaging over all pairs gives the marginal probability path \(\hl{pt}{p_t}\) and the marginal velocity field:

这里 \(p_t(\mathbf x\mid\mathbf z)\) 是集中在 \(\psi_t(\mathbf z)\) 一点上的分布；分式就是后验 \(p(\mathbf z\mid\mathbf x_t=\mathbf x)\)。一个速度场 \(\mathbf v\) “生成”一条概率路径，是指沿它漂流的样本在每个时刻的分布都是 \(p_t\)，等价地，二者满足连续性方程（质量守恒，3.6 节的散度在这里出现）：

Here \(p_t(\mathbf x\mid\mathbf z)\) is the distribution concentrated on the single point \(\psi_t(\mathbf z)\), and the fraction is the posterior \(p(\mathbf z\mid\mathbf x_t=\mathbf x)\). A velocity field \(\mathbf v\) "generates" a probability path if samples drifting along it are distributed as \(p_t\) at every time; equivalently, the two satisfy the continuity equation (conservation of mass, where the divergence of Lesson 3.6 appears):

右边那个形式沿着轨迹 \(\dot{\mathbf x}=\mathbf v_t(\mathbf x)\) 成立，它正是 19.3 节 FIG 5 上方的瞬时换元公式；把左式展开成 \(\partial_tp+\mathbf v\cdot\nabla p+p\,\nabla\cdot\mathbf v=0\)，前两项合起来就是沿轨迹的全导数。

定理 1（边缘场生成边缘路径）。每条条件路径都满足自己的连续性方程 \(\partial_tp_t(\mathbf x\mid\mathbf z)+\nabla\cdot\big(p_t(\mathbf x\mid\mathbf z)\,\mathbf u_t(\mathbf x\mid\mathbf z)\big)=0\)（一个点沿直线匀速移动，当然质量守恒）。对 \(\pi(\mathbf z)\) 积分，对时间和空间的导数都可以拿到积分外面：

The form on the right holds along trajectories \(\dot{\mathbf x}=\mathbf v_t(\mathbf x)\) and is exactly the instantaneous change of variables above FIG 5 of Lesson 19.3; expand the left form into \(\partial_tp+\mathbf v\cdot\nabla p+p\,\nabla\cdot\mathbf v=0\), and the first two terms together are the total derivative along a trajectory.

Theorem 1 (the marginal field generates the marginal path). Each conditional path satisfies its own continuity equation, \(\partial_tp_t(\mathbf x\mid\mathbf z)+\nabla\cdot\big(p_t(\mathbf x\mid\mathbf z)\,\mathbf u_t(\mathbf x\mid\mathbf z)\big)=0\) (a point moving along a straight line at constant speed obviously conserves mass). Integrate over \(\pi(\mathbf z)\); the time and space derivatives come out of the integral:

最后一步就是 \(\mathbf u_t\) 的定义（分子乘上 \(p_t\)）。所以 \(\mathbf u_t\) 满足 \(p_t\) 的连续性方程：从 \(p_0\) 出发沿 \(\mathbf u_t\) 积分，在 \(t=1\) 恰好得到 \(q\)。这就是 FIG 1 里紫色直线和蓝色曲线给出同一片底色的原因。

and the last step is just the definition of \(\mathbf u_t\) (multiply its numerator by \(p_t\)). So \(\mathbf u_t\) satisfies the continuity equation of \(p_t\): start from \(p_0\), integrate along \(\mathbf u_t\), and at \(t=1\) you get exactly \(q\). That is why the purple lines and the blue curves of FIG 1 produce the same shading.

理想的目标是 flow matching 损失 \(\mathcal L_{\mathrm{FM}}(\theta)=\mathbb E_{t,\,\mathbf x\sim p_t}\|\mathbf v_\theta(\mathbf x,t)-\mathbf u_t(\mathbf x)\|^2\)，可惜 \(\mathbf u_t\) 算不出来。定理 2（条件 flow matching）：\(\mathcal L_{\mathrm{CFM}}\) 与 \(\mathcal L_{\mathrm{FM}}\) 只差一个与 \(\theta\) 无关的常数，所以梯度相同。证明只需把两个平方展开：

The ideal objective is the flow matching loss \(\mathcal L_{\mathrm{FM}}(\theta)=\mathbb E_{t,\,\mathbf x\sim p_t}\|\mathbf v_\theta(\mathbf x,t)-\mathbf u_t(\mathbf x)\|^2\), but \(\mathbf u_t\) cannot be computed. Theorem 2 (conditional flow matching): \(\mathcal L_{\mathrm{CFM}}\) and \(\mathcal L_{\mathrm{FM}}\) differ only by a constant independent of \(\theta\), so their gradients are equal. The proof just expands the two squares:

第二行用了条件期望的塔性质（先固定 \(\mathbf x_t\) 对配对求平均）。最后一项就是 FIG 2 的地板 \(\mathcal L^*\)，它等于“给定 \(\mathbf x_t\) 时直线速度的方差”的期望。它在配对越确定时越小：FIG 4 里独立配对的损失停在 2.8 到 2.9 之间，OT 降到 0.25 左右，reflow 只有 0.01 左右，但三个网络的“好坏”并不能从这三个数比出来。Lipman 等的原始论文是对 \(\mathbf x_1\) 单独条件化，用高斯条件路径 \(p_t(\mathbf x\mid\mathbf x_1)=\mathcal N\big(t\mathbf x_1,(1-(1-\sigma_{\min})t)^2I\big)\)，条件速度 \(\big(\mathbf x_1-(1-\sigma_{\min})\mathbf x\big)/\big(1-(1-\sigma_{\min})t\big)\)；\(\sigma_{\min}\to0\) 时它就是本节的 \(\mathbf x_1-\mathbf x_0\)。

The second line uses the tower property of conditional expectation (fix \(\mathbf x_t\) and average over the pairs first). The last term is FIG 2's floor \(\mathcal L^*\), the expected variance of the line velocity given \(\mathbf x_t\). It shrinks as the pairing becomes more determined: in FIG 4 the loss of independent pairing stalls between 2.8 and 2.9, OT drops to about 0.25 and reflow to about 0.01, yet those three numbers say nothing about which network is better. The original paper by Lipman et al. conditions on \(\mathbf x_1\) alone, with the Gaussian conditional path \(p_t(\mathbf x\mid\mathbf x_1)=\mathcal N\big(t\mathbf x_1,(1-(1-\sigma_{\min})t)^2I\big)\) and conditional velocity \(\big(\mathbf x_1-(1-\sigma_{\min})\mathbf x\big)/\big(1-(1-\sigma_{\min})t\big)\); as \(\sigma_{\min}\to0\) this is the \(\mathbf x_1-\mathbf x_0\) of this lesson.

取一般的高斯路径 \(\mathbf x_t=\alpha_t\mathbf x_1+\sigma_t\mathbf x_0\)（直线路径 \(\alpha_t=t,\ \sigma_t=1-t\)；余弦路径 \(\alpha_t=\sin\tfrac{\pi t}{2},\ \sigma_t=\cos\tfrac{\pi t}{2}\)），条件速度 \(\dot\alpha_t\mathbf x_1+\dot\sigma_t\mathbf x_0\)。若数据是 \(q=\sum_k\pi_k\,\mathcal N(\boldsymbol\mu_k,s_k^2I)\)，给定分量 \(k\) 时 \(\mathbf x_t\) 与速度是联合高斯的，4.4 节的条件高斯公式直接给出：

Take a general Gaussian path \(\mathbf x_t=\alpha_t\mathbf x_1+\sigma_t\mathbf x_0\) (straight: \(\alpha_t=t,\ \sigma_t=1-t\); cosine: \(\alpha_t=\sin\tfrac{\pi t}{2},\ \sigma_t=\cos\tfrac{\pi t}{2}\)) with conditional velocity \(\dot\alpha_t\mathbf x_1+\dot\sigma_t\mathbf x_0\). If the data are \(q=\sum_k\pi_k\,\mathcal N(\boldsymbol\mu_k,s_k^2I)\), then given component \(k\), \(\mathbf x_t\) and the velocity are jointly Gaussian, and the conditional-Gaussian formula of Lesson 4.4 gives directly:

\(w_k\) 是 FIG 1 读数里的后验权重。代入直线路径和 \(t=0\)：\(c_k=-1\)、\(w_k=\pi_k\)，于是 \(\mathbf u_0(\mathbf x)=\mathbb E[\mathbf x_1]-\mathbf x\)，这就是 FIG 3 里“一步 Euler 落到均值”的来源。本页的双月和螺旋其实也是高斯混合（28 个和 26 个标准差 0.12 的小高斯沿曲线排开），所以 FIG 2 的场误差、FIG 3 的参考轨迹和 FIG 4 的 reflow 配对都能精确算出。

\(w_k\) is the posterior weight in FIG 1's readout. Plug in the straight path at \(t=0\): \(c_k=-1\) and \(w_k=\pi_k\), so \(\mathbf u_0(\mathbf x)=\mathbb E[\mathbf x_1]-\mathbf x\), which is where FIG 3's "one Euler step lands on the mean" comes from. The moons and the spiral on this page are in fact Gaussian mixtures too (28 and 26 small Gaussians with standard deviation 0.12 strung along curves), which is why FIG 2's field error, FIG 3's reference trajectories and FIG 4's reflow pairs can all be computed exactly.

19.6 节的 DDPM 用的也是一条高斯路径：信号系数 \(\alpha\) 与噪声系数 \(\sigma\) 满足 \(\alpha^2+\sigma^2=1\)（方差保持），只是时间方向相反（DDPM 的第 0 步是数据，见第 2 节的约定对照），网络预测的是噪声 \(\boldsymbol\varepsilon\)（就是这里的 \(\mathbf x_0\)）。在高斯路径上，知道了 \(\mathbf x_t\)，下面三个量中任意一个都能换算出另外两个，所以“预测速度”“预测噪声”“预测干净数据”携带的信息相同。对直线路径：

The DDPM of Lesson 19.6 also uses a Gaussian path: its signal coefficient \(\alpha\) and noise coefficient \(\sigma\) satisfy \(\alpha^2+\sigma^2=1\) (variance preserving), only time runs the other way (DDPM's step 0 is the data; see the convention note in Section 2), and the network predicts the noise \(\boldsymbol\varepsilon\) (the \(\mathbf x_0\) of this lesson). On a Gaussian path, once \(\mathbf x_t\) is known, any one of the three quantities below determines the other two, so "predict the velocity", "predict the noise" and "predict the clean data" carry the same information. For the straight path:

其中“帽子”都表示给定 \(\mathbf x_t\) 的条件均值，\(\hat{\mathbf v}=\mathbf u_t(\mathbf x_t)\)。第三个式子把速度和 19.6 节的得分（score）联系起来（Tweedie 公式）。19.6 节 DDIM 的确定性采样器，就是在离散地积分这类 ODE（Song 等 2021 称之为概率流 ODE）。所以 flow matching 与扩散不是两类不同的模型，而是同一个框架里的不同选择：哪条路径（\(\alpha_t,\sigma_t\)）、网络输出什么（这会改变损失在不同 \(t\) 上的权重）、怎样抽 \(t\)、以及用 ODE 还是 SDE 采样。直线路径的好处是目标最简单（\(\mathbf x_1-\mathbf x_0\)，处处有界，不需要调度表），并且在配对有序时轨迹真正变直。

where every hat denotes a conditional mean given \(\mathbf x_t\), and \(\hat{\mathbf v}=\mathbf u_t(\mathbf x_t)\). The third formula ties the velocity to the score of Lesson 19.6 (Tweedie's formula). The deterministic DDIM sampler of Lesson 19.6 integrates an ODE of exactly this kind in discrete steps (Song et al., 2021, call it the probability-flow ODE). So flow matching and diffusion are not two different kinds of model but different choices within one framework: which path (\(\alpha_t,\sigma_t\)), what the network outputs (which changes how the loss is weighted across \(t\)), how \(t\) is sampled, and whether sampling uses an ODE or an SDE. The straight path's merits are the simplest target (\(\mathbf x_1-\mathbf x_0\), bounded everywhere, no schedule) and trajectories that really become straight when the pairing is orderly.

Euler 法每步的局部误差是 \(O(h^2)\)，累积 \(1/h\) 步后，与精确的流映射 \(\hl{ode}{\varphi_1(\mathbf x_0)}\)（3.6 节）之间的全局误差是 \(O(h)\)；中点法是 \(O(h^2)\)（2.5 节）。误差的常数取决于轨迹有多弯：如果每条轨迹都是匀速直线，\(\mathbf v\) 沿轨迹不变，Euler 一步就精确，\(\hl{euler}{\mathbf x_1=\mathbf x_0+\mathbf v_\theta(\mathbf x_0,0)}\)。什么样的配对能做到？Benamou 与 Brenier（2000）证明，二次代价的最优传输可以写成“动能最小的流”：

Euler's method has a local error of \(O(h^2)\) per step and, after \(1/h\) steps, a global error of \(O(h)\) relative to the exact flow map \(\hl{ode}{\varphi_1(\mathbf x_0)}\) (Lesson 3.6); the midpoint method's is \(O(h^2)\) (Lesson 2.5). The constant depends on how curved the trajectories are: if every trajectory is a straight line traversed at constant speed, \(\mathbf v\) is constant along it and a single Euler step is exact, \(\hl{euler}{\mathbf x_1=\mathbf x_0+\mathbf v_\theta(\mathbf x_0,0)}\). Which pairing achieves this? Benamou and Brenier (2000) showed that optimal transport with quadratic cost can be written as "the flow of least kinetic energy":

最优的流沿直线匀速搬运，粒子互不相撞。所以用最优传输配对再走直线，边缘轨迹就是直线（FIG 4 右边那些青色的配对线段 \(\hl{pair}{(\mathbf x_0,\mathbf x_1)}\) 短而几乎不交叉）。小批量 OT 只在每个批次内部求最优，是近似，批次越大越接近；reflow 用一个已有的 ODE 映射作配对，Liu 等证明这样做不会增加任何凸代价下的传输代价，并且反复 reflow 会让轨迹越来越直。代价是要多训练一次，并且配对来自模型本身，模型的误差会被带进新的数据里。

The optimal flow carries mass along straight lines at constant speed, and particles never collide. So pair by optimal transport, walk straight lines, and the marginal trajectories are straight (the cyan pair segments \(\hl{pair}{(\mathbf x_0,\mathbf x_1)}\) on the right of FIG 4 are short and hardly cross). Minibatch OT optimizes only within each batch, an approximation that improves with batch size; reflow uses an existing ODE map as the pairing, which Liu et al. proved never increases the transport cost under any convex cost, and repeated reflow makes the trajectories straighter and straighter. The price is another round of training, and the pairs come from the model itself, so its errors are baked into the new data.

条件生成：把条件 \(\mathbf c\)（文本、图像特征、机器人观测）作为网络的额外输入，\(\mathbf v_\theta(\mathbf x,t,\mathbf c)\)，训练时配对取自同一个 \(\mathbf c\) 下的数据，其余不变；第 6 节的动作头就是这样。无分类器引导（19.6 节）原样适用于速度：\(\tilde{\mathbf v}=\mathbf v_\theta(\mathbf x,t,\varnothing)+w\,\big(\mathbf v_\theta(\mathbf x,t,\mathbf c)-\mathbf v_\theta(\mathbf x,t,\varnothing)\big)\)。似然：训练好的流是一个连续 normalizing flow，19.3 节的公式 \(\log p_1(\mathbf x_1)=\log p_0(\mathbf x_0)-\int_0^1\nabla\cdot\mathbf v_\theta\,dt\) 照样成立，需要时可以倒着积分求密度。时间的抽样：本页用 \(t\sim\mathcal U[0,1]\)；Stable Diffusion 3（Esser 等，2024）用 logit-正态分布把训练集中在中间的 \(t\)，π0 用一个偏向噪声一端的 Beta 型分布。这只改变损失在不同时刻上的权重，不改变最优解 \(\mathbf u_t\)。

Conditional generation: feed the condition \(\mathbf c\) (text, image features, robot observations) to the network as an extra input, \(\mathbf v_\theta(\mathbf x,t,\mathbf c)\), and draw each training pair from data under the same \(\mathbf c\); nothing else changes. The action head of Section 6 does exactly this. Classifier-free guidance (Lesson 19.6) carries over to velocities as is: \(\tilde{\mathbf v}=\mathbf v_\theta(\mathbf x,t,\varnothing)+w\,\big(\mathbf v_\theta(\mathbf x,t,\mathbf c)-\mathbf v_\theta(\mathbf x,t,\varnothing)\big)\). Likelihood: the trained flow is a continuous normalizing flow, so the formula of Lesson 19.3, \(\log p_1(\mathbf x_1)=\log p_0(\mathbf x_0)-\int_0^1\nabla\cdot\mathbf v_\theta\,dt\), still holds, and densities can be computed by integrating backwards when needed. Sampling the time: this page uses \(t\sim\mathcal U[0,1]\); Stable Diffusion 3 (Esser et al., 2024) uses a logit-normal distribution that concentrates training on intermediate \(t\), and π0 uses a Beta-shaped distribution that leans towards the noisy end. This only changes how the loss is weighted over time, not the optimum \(\mathbf u_t\).

π0（Black 等，Physical Intelligence，2024）是一个视觉-语言-动作模型：一个预训练的视觉-语言模型（PaliGemma）读入多路相机图像、语言指令和机器人本体状态，旁边接一个约 3 亿参数的动作专家，用 flow matching 生成一整个动作块（未来 50 步的连续动作）。推理时从高斯噪声出发积分 10 步（步长 0.1）；观测部分的注意力键值只算一次并缓存，10 步里只重复运行较小的动作专家。这样它能以高达 50 Hz 的频率控制双臂等机器人完成叠衣服这类灵巧任务。与 17.5 节的离散动作 token 相比，这条路线直接输出连续动作，不需要动作 tokenizer。

下图是一个能在浏览器里训练的缩小版。一个点机器人以恒定速度从左向右开，经过一个圆形障碍物（半径 0.55），观测 \(\hl{obs}{\mathbf o=(y_s,y_o)}\) 是起点高度和障碍物高度，动作块是未来 \(H=8\) 步的横向位置（相对起点）。示教数据里，人从障碍物上方绕过的概率是 \(\sigma(-6y_o)\)：障碍物越靠上，越多人从下方走。速度网络 \(\mathbf v_\theta(\mathbf a_t,t,\mathbf o)\) 是 \([13,64,64,8]\) 的 GELU MLP（8 维动作 + 2 维观测 + 3 个时间特征），共 5576 个参数；作为对照，一个 \([2,64,64,8]\) 的网络直接用均方误差回归动作块，也就是普通的行为克隆（22.1 节）。两者同时训练，Adam，每步 64 条示教。

19.6 节的 FIG 6 已经在一个固定的避障场景里，用精确的 score 对整段动作去噪，展示了每段动作都完整地选定一边、两边的频率与示教一致。这里不再重复那一点，而是看三样新东西：速度场是网络现场学出来的；它以观测为条件，障碍物一动，两种走法的比例就跟着变；积分步数可以一直减到 1，这时 flow matching 退化成均方误差回归。

π0 (Black et al., Physical Intelligence, 2024) is a vision-language-action model: a pretrained vision-language model (PaliGemma) reads several camera images, a language instruction and the robot's proprioceptive state, and next to it sits an action expert of about 300 million parameters that generates a whole action chunk (the next 50 continuous actions) by flow matching. At inference it integrates 10 steps (step size 0.1) from Gaussian noise; the attention keys and values of the observation part are computed once and cached, so only the smaller action expert reruns during the 10 steps. This lets it control robots such as bimanual arms at up to 50 Hz in dexterous tasks like folding laundry. Compared with the discrete action tokens of Lesson 17.5, this route outputs continuous actions directly and needs no action tokenizer.

The figure below is a scaled-down version you can train in the browser. A point robot drives from left to right at constant speed past a disc obstacle (radius 0.55); the observation \(\hl{obs}{\mathbf o=(y_s,y_o)}\) is the start height and the obstacle height, and the action chunk is the lateral position at the next \(H=8\) steps (relative to the start). In the demonstrations a person passes above the obstacle with probability \(\sigma(-6y_o)\): the higher the obstacle, the more people go below it. The velocity network \(\mathbf v_\theta(\mathbf a_t,t,\mathbf o)\) is a \([13,64,64,8]\) GELU MLP (8 action dimensions + 2 observation dimensions + 3 time features), 5576 parameters; for comparison, a \([2,64,64,8]\) network regresses the action chunk directly with a mean squared error, i.e. plain behaviour cloning (Lesson 22.1). Both train together with Adam on 64 demonstrations per step.

FIG 6 of Lesson 19.6 already denoised whole action chunks around an obstacle in one fixed scene with the exact score, and showed that each chunk commits fully to one side, with the two sides at the demonstrated frequencies. Rather than repeat that, this figure adds three new things: the velocity field is learned live by a network; it is conditioned on the observation, so the ratio of the two ways round changes as the obstacle moves; and the number of integration steps can go all the way down to 1, where flow matching collapses into mean-squared-error regression.