Normalizing Flow（归一化流）Normalizing Flows

机器人经常需要的不是“一个答案”，而是“答案的分布”。一个 7 自由度机械臂够到同一个杯子有无数种姿态；同一张桌面能有很多个好的抓取位姿；示教数据里，同一个场景下人有时从左绕、有时从右绕。我们想要一个模型，能做两件事：采样（给我 50 个不同的可行姿态），以及打分（这个姿态在数据里有多常见，\(p_\theta(\mathbf x)\) 是多少）。打分还能用来做异常检测：读数的似然低得离谱，说明遇到了没见过的情况。

前两节的生成模型各缺一半。19.1 节的 VAE 能采样，但只给出似然的一个下界（ELBO）；19.2 节的 GAN 采样质量好，却完全没有密度可算。Normalizing flow 的做法很直接：拿一个能精确算密度的分布（标准高斯 \(p_Z\)），再用一个可逆的神经网络 \(f\) 把它变形成数据的样子。采样就是 \(\mathbf x=f(\mathbf z)\)；打分就是把 \(\mathbf x\) 倒推回 \(\mathbf z=f^{-1}(\mathbf x)\)，读出那里的高斯密度，再按 \(f\) 把体积放大了多少倍做修正。这个“修正”就是 1.5 节说过的那句话：变换之后，密度要除以 \(|\det|\)。

本节的关键问题是工程上的：一般的神经网络既不可逆，雅可比行列式也要 \(O(D^3)\) 才能算。RealNVP 的耦合层用一个巧妙的结构同时解决了两个问题，于是可以直接用最大似然（4.6 节）训练。在机器人里，这让 flow 成了逆运动学采样器、策略分布和位姿不确定性模型的一个选项；它的连续时间版本则直接引出了今天机器人策略里常用的 flow matching（19.7 节）。

Often a robot does not need "an answer" but "the distribution of answers". A 7-DoF arm can reach the same cup in infinitely many postures; one tabletop admits many good grasp poses; in demonstration data a person sometimes walks around an obstacle on the left and sometimes on the right. We want a model that does two things: sample (give me 50 different feasible postures) and score (how common is this posture in the data, what is \(p_\theta(\mathbf x)\)). Scoring also gives anomaly detection: an absurdly low likelihood means the robot is seeing something new.

The previous two lessons' generative models each miss one half. The VAE of Lesson 19.1 samples, but only gives a lower bound on the likelihood (the ELBO); the GAN of Lesson 19.2 produces good samples but has no density at all. A normalizing flow takes the direct route: start from a distribution whose density is known exactly (the standard Gaussian \(p_Z\)) and deform it into the shape of the data with an invertible neural network \(f\). Sampling is \(\mathbf x=f(\mathbf z)\); scoring runs \(\mathbf x\) backwards to \(\mathbf z=f^{-1}(\mathbf x)\), reads the Gaussian density there, and corrects for how much \(f\) magnified volume. That correction is the sentence from Lesson 1.5: after a transformation, the density must be divided by \(|\det|\).

The key problem of this lesson is an engineering one: a generic neural network is not invertible, and its Jacobian determinant costs \(O(D^3)\). The coupling layer of RealNVP solves both at once with a clever structure, so the model can be trained directly by maximum likelihood (Lesson 4.6). In robotics this makes flows an option for inverse-kinematics samplers, policy distributions and pose-uncertainty models, and their continuous-time version leads straight to flow matching (Lesson 19.7), which today's robot policies use widely.

先看一维。\(z\) 服从标准正态，再取一个单调递增的函数 \(x=f(z)\)。把 \(z\) 轴想成一条橡皮筋，上面均匀撒着“概率沙子”，\(f\) 把橡皮筋的各段拉长或压短。沙子总量不变：落在 \([z,z+dz]\) 里的沙子，变换后全部落在 \([f(z),f(z)+f'(z)\,dz]\) 里。所以那一段被拉长 \(f'(z)\) 倍，沙子的密度就变为原来的 \(1/f'(z)\)：

Start in one dimension. Let \(z\) be standard normal and take a monotonically increasing function \(x=f(z)\). Think of the \(z\) axis as a rubber band sprinkled with "probability sand"; \(f\) stretches some pieces of the band and squeezes others. The amount of sand does not change: the sand in \([z,z+dz]\) all lands in \([f(z),f(z)+f'(z)\,dz]\) after the map. So if a piece is stretched by a factor \(f'(z)\), the density of the sand there becomes \(1/f'(z)\) of what it was:

下图的 \(f\) 是一条折线，经过 7 个节点 \(z=-3,-2,\dots,3\)。你可以拖动每个节点在下方 \(x\) 轴上的落点（保持单调）。每一段的斜率就是 \(f'\)：段被拉长（斜率 > 1）的地方密度变低，被压扁（斜率 < 1）的地方密度变高。4.9 节的逆 CDF 采样其实就是一个一维 flow：它把均匀分布捏成目标分布，用的正是这个公式。

In the figure below \(f\) is a broken line through 7 knots at \(z=-3,-2,\dots,3\). Drag where each knot lands on the lower \(x\) axis (order is kept). The slope of each piece is \(f'\): where a piece is stretched (slope > 1) the density drops, where it is squeezed (slope < 1) the density rises. The inverse-CDF sampling of Lesson 4.9 is in fact a 1-D flow: it moulds the uniform distribution into the target with exactly this formula.

二维时，flow 需要一个可逆的 \(f:\R^2\to\R^2\)，并且要能便宜地算出 \(\det J_f\)。RealNVP（Dinh 等，2017）的仿射耦合层是这样的：把坐标分成两半，第一半原样通过，第二半被缩放再平移，而缩放量 \(\hl{s}{s}\) 和平移量 \(\hl{t}{t}\) 由第一半算出来（可以是任意神经网络）：

In two dimensions a flow needs an invertible \(f:\R^2\to\R^2\) with a cheap \(\det J_f\). The affine coupling layer of RealNVP (Dinh et al., 2017) works like this: split the coordinates into two halves; the first half passes through unchanged, the second half is scaled and then shifted, and the scale \(\hl{s}{s}\) and shift \(\hl{t}{t}\) are computed from the first half (by any neural network you like):

两个好处一眼可见。可逆：因为 \(z_1=x_1\) 原样保留，倒推时先拿到 \(z_1\)，就能重新算出 \(s\) 和 \(t\)，再把缩放平移撤销；\(s\)、\(t\) 本身完全不需要可逆。行列式便宜：\(x_1\) 不依赖 \(z_2\)，所以雅可比是下三角的，行列式就是对角线的乘积 \(1\cdot e^{s(z_1)}\)，不管 \(s\) 和 \(t\) 多复杂。下图用两个手写的函数 \(s(z_1)=\alpha\sin(1.3z_1)\)、\(t(z_1)=\beta(\tfrac12z_1^2-0.8)\) 代替神经网络，让你看清一层到底做了什么。

Two benefits are visible at once. Invertible: since \(z_1=x_1\) is kept as is, running backwards you get \(z_1\) first, recompute \(s\) and \(t\) from it, and undo the scale and shift; \(s\) and \(t\) themselves need not be invertible at all. Cheap determinant: \(x_1\) does not depend on \(z_2\), so the Jacobian is lower triangular and its determinant is the product of the diagonal, \(1\cdot e^{s(z_1)}\), however complicated \(s\) and \(t\) are. The figure below replaces the networks with two hand-written functions, \(s(z_1)=\alpha\sin(1.3z_1)\) and \(t(z_1)=\beta(\tfrac12z_1^2-0.8)\), so you can see exactly what one layer does.

现在把 \(s\) 和 \(t\) 换成小神经网络（每层一个 \([1,16,16,2]\) 的 tanh MLP，同时输出 \(s\) 和 \(t\)），叠 8 层，两半坐标轮流被变换，一共 2704 个参数。训练目标就是 4.6 节的最大似然：对每个数据点倒推到 \(\mathbf z\)，累加每层的对数行列式，最小化平均负对数似然（NLL，单位 nat）：

Now replace \(s\) and \(t\) by small neural networks (one \([1,16,16,2]\) tanh MLP per layer, outputting \(s\) and \(t\) together), stack 8 layers with the two halves taking turns being transformed: 2704 parameters in all. The training objective is the maximum likelihood of Lesson 4.6: run each data point back to \(\mathbf z\), add up every layer's log-determinant, and minimize the mean negative log-likelihood (NLL, in nats):

第一项 \(\tfrac12\|\mathbf z\|^2\) 要求数据被推到高斯的中心，单独用它，网络会把所有点压成一个点（逆变换的缩放 \(e^{-s}\to0\)，即 \(s\to+\infty\)）；第二项 \(\sum s\) 惩罚这种压缩，因为压缩就是 \(\det J_{f^{-1}}\) 很小。两项拉锯的结果是：把数据均匀地铺满一个标准高斯。梯度用 16.3 节的反向传播逐层算出（页面里与有限差分对过，误差约 \(10^{-9}\)），优化器是 Adam（学习率 0.005，2000 步后降到 0.002），每步 64 个样本。

The first term \(\tfrac12\|\mathbf z\|^2\) wants the data pushed to the centre of the Gaussian; on its own it would let the network crush every point into one (the inverse scale \(e^{-s}\to0\), that is \(s\to+\infty\)). The second term \(\sum s\) penalizes exactly that crushing, because crushing means a tiny \(\det J_{f^{-1}}\). The tug of war between them ends with the data spread evenly over a standard Gaussian. The gradients come from the backpropagation of Lesson 16.3, layer by layer (checked against finite differences while writing this page, error about \(10^{-9}\)); the optimizer is Adam (learning rate 0.005, dropped to 0.002 after 2000 steps) on mini-batches of 64.

“Normalizing”这个名字来自密度估计的方向：从数据 \(\mathbf x\) 出发，一层一层倒推，直到分布变成标准正态（normal）。下图取 FIG 3 正在训练的那个模型，把 500 个测试点逐层推过去，按它们来自哪条月牙（或哪个团）着色。反过来选“\(\mathbf z\to\) 数据”，看到的就是采样：一团高斯一层层被捏成数据的形状。

The name "normalizing" comes from the density-estimation direction: start from the data \(\mathbf x\) and undo the layers one by one until the distribution becomes standard normal. The figure below takes the very model FIG 3 is training and pushes the 500 test points through it layer by layer, coloured by which moon (or blob) they came from. Choose "\(\mathbf z\to\) data" instead and you see sampling: a Gaussian cloud being moulded layer by layer into the shape of the data.

注意 \(\mathbf z\) 端的颜色：两类数据被分到高斯的两边，中间只有一条细缝。flow 学到的 \(\mathbf z\) 是数据的一种坐标系，和数据同维度、一一对应，不像 VAE 的隐变量那样可以更低维、并带着噪声（19.1 节）。按默认种子训练到 4000 步时，平均 \(\tfrac12\|\mathbf z\|^2\) 约为 1.16，略大于 1：模型还没把数据完全“高斯化”。

Look at the colours at the \(\mathbf z\) end: the two classes are sent to two sides of the Gaussian with only a thin seam between them. The \(\mathbf z\) a flow learns is a coordinate system for the data, of the same dimension and one-to-one, unlike a VAE's latent, which can be lower-dimensional and noisy (Lesson 19.1). With the default seed, after 4000 steps the mean \(\tfrac12\|\mathbf z\|^2\) is about 1.16, a little above 1: the model has not quite "Gaussianized" the data yet.

如果每一层只做一个极小的变形 \(\mathbf x\leftarrow\mathbf x+\Delta t\,\mathbf v(\mathbf x)\)，叠无穷多层，flow 就变成了一个常微分方程（2.5 节）：每个样本沿着向量场 \(\hl{vf}{\mathbf v(\mathbf x,t)}\) 漂流（3.6 节）。这时“对数行列式之和”变成了沿轨迹对散度的积分：一小块区域的面积按 \(\tfrac{d}{dt}\log A=\hl{div}{\nabla\cdot\mathbf v}\) 变化（3.6 节说过：散度就是面积的增长率），密度则按相反的符号变化。这就是 Neural ODE（Chen 等，2018）提出的瞬时换元公式。下图用三个解析的向量场演示它；底色是散度。

If each layer makes only a tiny deformation \(\mathbf x\leftarrow\mathbf x+\Delta t\,\mathbf v(\mathbf x)\) and we stack infinitely many, the flow becomes an ordinary differential equation (Lesson 2.5): every sample drifts along a vector field \(\hl{vf}{\mathbf v(\mathbf x,t)}\) (Lesson 3.6). The "sum of log-determinants" then becomes an integral of the divergence along the trajectory: a small patch's area changes as \(\tfrac{d}{dt}\log A=\hl{div}{\nabla\cdot\mathbf v}\) (Lesson 3.6: divergence is the growth rate of area), and the density changes with the opposite sign. This is the instantaneous change of variables introduced with Neural ODEs (Chen et al., 2018). The figure demonstrates it with three analytic vector fields; the background shading is the divergence.

设 \(f:\R^D\to\R^D\) 是可逆且可微的（微分同胚），\(\mathbf z\sim p_Z\)，\(\mathbf x=f(\mathbf z)\)。对任意区域 \(\Omega\)，\(\mathbf x\) 落在 \(f(\Omega)\) 的概率等于 \(\mathbf z\) 落在 \(\Omega\) 的概率；而 3.3 节告诉我们，\(f\) 把 \(\mathbf z\) 附近的小体积放大 \(|\det J_f(\mathbf z)|\) 倍。于是

Let \(f:\R^D\to\R^D\) be invertible and differentiable (a diffeomorphism), \(\mathbf z\sim p_Z\), \(\mathbf x=f(\mathbf z)\). For any region \(\Omega\), the probability that \(\mathbf x\) lands in \(f(\Omega)\) equals the probability that \(\mathbf z\) lands in \(\Omega\), and Lesson 3.3 tells us \(f\) magnifies small volumes near \(\mathbf z\) by \(|\det J_f(\mathbf z)|\). Hence

两种写法等价，因为反函数的雅可比是雅可比的逆（反函数定理：\(J_{f^{-1}}(\mathbf x)=J_f(\mathbf z)^{-1}\)），行列式取倒数。把 \(f\) 写成 \(K\) 层的复合 \(f=f_K\circ\cdots\circ f_1\)，中间量 \(\mathbf h_0=\mathbf z,\ \mathbf h_k=f_k(\mathbf h_{k-1}),\ \mathbf h_K=\mathbf x\)。链式法则让雅可比相乘，行列式也相乘，取对数后变成求和：

The two forms are equivalent because the Jacobian of the inverse is the inverse of the Jacobian (the inverse function theorem: \(J_{f^{-1}}(\mathbf x)=J_f(\mathbf z)^{-1}\)), whose determinant is the reciprocal. Write \(f\) as a composition of \(K\) layers, \(f=f_K\circ\cdots\circ f_1\), with intermediates \(\mathbf h_0=\mathbf z,\ \mathbf h_k=f_k(\mathbf h_{k-1}),\ \mathbf h_K=\mathbf x\). The chain rule multiplies the Jacobians, so the determinants multiply too, and after taking logs they add:

这就是 FIG 4 读数里的那笔账：在 \(\mathbf z\) 端，平均 \(\tfrac12\|\mathbf z\|^2\) 加上 \(\tfrac D2\log2\pi\)（\(D=2\) 时就是 \(\log2\pi\)），减去累积的平均 \(\log|\det J_{f^{-1}}|\)，正好是测试 NLL。

This is the bookkeeping in FIG 4's readout: at the \(\mathbf z\) end, the mean \(\tfrac12\|\mathbf z\|^2\) plus \(\tfrac D2\log2\pi\) (just \(\log2\pi\) for \(D=2\)) minus the accumulated mean \(\log|\det J_{f^{-1}}|\) is exactly the test NLL.

训练目标 \(\hl{nll}{L(\theta)}=-\tfrac1N\sum_i\log p_\theta(\mathbf x_i)\) 是期望 \(-\mathbb E_{p_{\mathcal D}}[\log p_\theta]\) 的样本估计，即交叉熵 \(H(p_{\mathcal D},p_\theta)=H(p_{\mathcal D})+\mathrm{KL}(p_{\mathcal D}\|p_\theta)\)（4.11 节）。熵 \(H(p_{\mathcal D})\) 与 \(\theta\) 无关，所以最小化 NLL 就是最小化 KL，并且 NLL 永远不低于 \(H(p_{\mathcal D})\)：这就是 FIG 3 那条绿色虚线。两者之差就是模型离真实分布还有多远，以 nat 计。注意 KL 的这个方向（数据在前）惩罚的是“数据有、模型没有”，所以最大似然训练的模型倾向于覆盖所有模式，宁可在模式之间留一些多余的密度，这和 GAN 容易丢模式（19.2 节）正好相反。

The objective \(\hl{nll}{L(\theta)}=-\tfrac1N\sum_i\log p_\theta(\mathbf x_i)\) is a sample estimate of \(-\mathbb E_{p_{\mathcal D}}[\log p_\theta]\), the cross-entropy \(H(p_{\mathcal D},p_\theta)=H(p_{\mathcal D})+\mathrm{KL}(p_{\mathcal D}\|p_\theta)\) (Lesson 4.11). The entropy \(H(p_{\mathcal D})\) does not depend on \(\theta\), so minimizing the NLL is minimizing the KL, and the NLL can never go below \(H(p_{\mathcal D})\): that is the green dashed line of FIG 3. The gap between them is how far the model still is from the truth, in nats. Note that this direction of KL (data first) penalizes "the data has it, the model does not", so maximum-likelihood models tend to cover every mode, leaving some extra density between modes rather than dropping one, the opposite of the GAN's tendency to drop modes (Lesson 19.2).

一般地，把 \(\mathbf z\) 分成 \((\mathbf z_a,\mathbf z_b)\)，仿射耦合层是 \(\mathbf x_a=\mathbf z_a\)，\(\mathbf x_b=\mathbf z_b\odot e^{\hl{s}{\mathbf s(\mathbf z_a)}}+\hl{t}{\mathbf t(\mathbf z_a)}\)（\(\odot\) 是逐元素乘）。它的雅可比是

In general, split \(\mathbf z\) into \((\mathbf z_a,\mathbf z_b)\); the affine coupling layer is \(\mathbf x_a=\mathbf z_a\), \(\mathbf x_b=\mathbf z_b\odot e^{\hl{s}{\mathbf s(\mathbf z_a)}}+\hl{t}{\mathbf t(\mathbf z_a)}\) (\(\odot\) is the elementwise product). Its Jacobian is

左下块 \(\partial\mathbf x_b/\partial\mathbf z_a\) 牵涉 \(\mathbf s\) 和 \(\mathbf t\) 网络的导数，可能非常复杂，但三角矩阵的行列式只看对角线，所以根本用不着它。这就是全部窍门：网络的表达能力放在 \(\mathbf s,\mathbf t\) 里，可逆性和行列式由结构保证。代价是每层只变换一半坐标，所以必须轮换哪一半被变换（FIG 3 的“总是变换 \(x_2\)”就是反例）；高维时在层之间插入固定的置换，Glow（Kingma 和 Dhariwal，2018）则学一个可逆的 \(1\times1\) 卷积来混合通道，用 LU 分解让它的行列式也便宜。

成本对比。一般的 \(D\times D\) 雅可比求行列式要 \(O(D^3)\) 次运算；耦合层只要 \(O(D)\)，加上一次网络前向。对一张 \(64\times64\) 的 RGB 图像，\(D=12288\)，\(D^3/3\approx6\times10^{11}\)：每个样本、每一层都要这么多次运算，是不可能用于训练的。

其他结构。自回归 flow（MAF，Papamakarios 等，2017；IAF，Kingma 等，2016）让第 \(i\) 个坐标的 \(s_i,t_i\) 依赖前面所有坐标，雅可比同样是三角的，但两个方向里有一个必须逐坐标串行计算：MAF 算密度快、采样慢，IAF 反之（19.4 节的自回归模型是同一种因式分解）。样条 flow（Durkan 等，2019）把仿射的 \(e^sz+t\) 换成单调的有理二次样条，表达力更强，FIG 1 的折线就是它最简陋的版本。

The lower-left block \(\partial\mathbf x_b/\partial\mathbf z_a\) involves the derivatives of the \(\mathbf s\) and \(\mathbf t\) networks and may be very complicated, but the determinant of a triangular matrix only looks at the diagonal, so it is never needed. That is the whole trick: the expressive power lives in \(\mathbf s,\mathbf t\), while invertibility and the determinant are guaranteed by the structure. The price is that each layer transforms only half the coordinates, so the transformed half must alternate (FIG 3's "always transform \(x_2\)" is the counterexample); in high dimensions fixed permutations are inserted between layers, and Glow (Kingma and Dhariwal, 2018) learns an invertible \(1\times1\) convolution to mix channels, LU-factorized so that its determinant is cheap too.

Cost comparison. The determinant of a generic \(D\times D\) Jacobian takes \(O(D^3)\) operations; a coupling layer takes \(O(D)\) plus one network pass. For a \(64\times64\) RGB image \(D=12288\) and \(D^3/3\approx6\times10^{11}\) operations, for every sample and every layer: hopeless for training.

Other designs. Autoregressive flows (MAF, Papamakarios et al., 2017; IAF, Kingma et al., 2016) let the \(s_i,t_i\) of coordinate \(i\) depend on all earlier coordinates; the Jacobian is again triangular, but one of the two directions must run coordinate by coordinate: MAF evaluates densities fast and samples slowly, IAF the reverse (the autoregressive models of Lesson 19.4 use the same factorization). Spline flows (Durkan et al., 2019) replace the affine \(e^sz+t\) with monotone rational-quadratic splines for more expressive power; FIG 1's broken line is their crudest version.

训练时走的是密度方向：每层 \(w_b=(u_b-t)e^{-s}\)，损失 \(\ell=\tfrac12\|\mathbf z\|^2+\sum s\)。设 \(g=\partial\ell/\partial w_b\)（最后一层 \(g=z_b\)），反向传播一层需要的三个局部导数是

Training runs in the density direction: each layer computes \(w_b=(u_b-t)e^{-s}\) and the loss is \(\ell=\tfrac12\|\mathbf z\|^2+\sum s\). With \(g=\partial\ell/\partial w_b\) (at the last layer \(g=z_b\)), backpropagating through one layer needs three local derivatives:

后两个再经网络反传到 \(\mathbf u_a\)，与直通的 \(\partial\ell/\partial w_a\) 相加。注意 \(\partial\ell/\partial s\) 里的 “1”：它来自对数行列式，没有它就没有阻止塌缩的力。页面里的 \(s\) 是 \(2\tanh(\cdot/2)\) 限幅后的网络输出，防止 \(e^{s}\) 溢出（误区 5）。

The last two are backpropagated through the network to \(\mathbf u_a\) and added to the pass-through \(\partial\ell/\partial w_a\). Notice the "1" in \(\partial\ell/\partial s\): it comes from the log-determinant, and without it nothing would resist collapse. On this page \(s\) is the network output squashed by \(2\tanh(\cdot/2)\), so \(e^{s}\) cannot overflow (Pitfall 5).

把一层 \(\mathbf x\leftarrow\mathbf x+\Delta t\,\mathbf v\) 的雅可比写成 \(I+\Delta t\,\partial\mathbf v/\partial\mathbf x\)，则 \(\log\det(I+\Delta t A)=\Delta t\operatorname{tr}A+O(\Delta t^2)\)，而 \(\operatorname{tr}(\partial\mathbf v/\partial\mathbf x)\) 正是散度 \(\hl{div}{\nabla\cdot\mathbf v}\)。让 \(\Delta t\to0\)，就得到 FIG 5 上方那个公式。连续 flow 的好处是 \(\mathbf v\) 可以是任意神经网络，不需要特殊结构就自动可逆（ODE 的解在合理条件下唯一，倒着积分就是逆）。坏处是按最大似然训练它要在每一步都把 ODE 积分一遍，还要算散度；FFJORD（Grathwohl 等，2019）用 Hutchinson 随机迹估计降低了散度的成本，但积分仍然昂贵。19.7 节的 flow matching 换了一条路：不再最大化似然，而是直接对一个事先选好的概率路径上的速度场做回归，训练时完全不用积分 ODE。

Write the Jacobian of one layer \(\mathbf x\leftarrow\mathbf x+\Delta t\,\mathbf v\) as \(I+\Delta t\,\partial\mathbf v/\partial\mathbf x\). Then \(\log\det(I+\Delta t A)=\Delta t\operatorname{tr}A+O(\Delta t^2)\), and \(\operatorname{tr}(\partial\mathbf v/\partial\mathbf x)\) is exactly the divergence \(\hl{div}{\nabla\cdot\mathbf v}\). Let \(\Delta t\to0\) and you get the formula above FIG 5. The appeal of continuous flows is that \(\mathbf v\) can be any neural network: no special structure is needed for invertibility (under mild conditions the ODE solution is unique, and integrating backwards inverts it). The drawback is that maximum-likelihood training must integrate the ODE at every step and compute the divergence along the way; FFJORD (Grathwohl et al., 2019) cut the cost of the divergence with Hutchinson's stochastic trace estimator, but the integration remains expensive. Flow matching in Lesson 19.7 takes another road: instead of maximizing likelihood it regresses the velocity field of a probability path chosen in advance, so training never integrates the ODE at all.

冗余机械臂的逆运动学（7.5 节）有无穷多个解。数值 IK 每次从一个初值出发收敛到一个解；想要很多个不同的解（比如避障时挑一个离障碍最远的），只能换很多初值反复跑。条件 flow 换个思路：学一个分布 \(p_\theta(\mathbf q\mid\hl{cond}{\mathbf c})\)，给定末端目标 \(\mathbf c\)，一次就采出一批关节角。条件的做法很简单：把 \(\mathbf c\) 拼进每个耦合层 \(\mathbf s,\mathbf t\) 网络的输入，\(\mathbf s(\mathbf q_a,\mathbf c)\)、\(\mathbf t(\mathbf q_a,\mathbf c)\)；可逆性只针对 \(\mathbf q\)，所以三角结构照旧成立。

训练数据来自正运动学，几乎不要钱：随机采关节角 \(q_i\sim\mathcal N(0,0.9^2)\)，算出末端位置，加上 0.04 m 的噪声作为 \(\mathbf c\)。下图是一个三连杆平面臂（杆长 1.0、0.8、0.6 m），目标是二维位置，所以每个目标对应一条一维的解曲线（自运动流形）。模型有 6 个耦合层（每层只变换一个关节角，三个关节轮流），每层一个 \([4,24,24,2]\) 的网络，共 4620 个参数。

The inverse kinematics of a redundant arm (Lesson 7.5) has infinitely many solutions. Numerical IK converges from one start to one solution; if you want many different ones (say, to pick the one farthest from an obstacle) you must rerun it from many starts. A conditional flow does it differently: learn a distribution \(p_\theta(\mathbf q\mid\hl{cond}{\mathbf c})\) and, given an end-effector target \(\mathbf c\), sample a whole batch of joint angles in one go. Conditioning is simple: append \(\mathbf c\) to the input of every coupling layer's networks, \(\mathbf s(\mathbf q_a,\mathbf c)\) and \(\mathbf t(\mathbf q_a,\mathbf c)\); invertibility is only needed in \(\mathbf q\), so the triangular structure still holds.

Training data comes from forward kinematics almost for free: sample joint angles \(q_i\sim\mathcal N(0,0.9^2)\), compute the tip position, add 0.04 m of noise and call it \(\mathbf c\). The figure uses a planar three-link arm (links 1.0, 0.8 and 0.6 m) with a 2-D position target, so each target has a one-dimensional curve of solutions (the self-motion manifold). The model has 6 coupling layers (each transforms one joint angle, the three joints taking turns), each with a \([4,24,24,2]\) network: 4620 parameters.