深度学习（生成式模型）——score-based generative modeling through stochastic differential equations

前言

yang song博士在《Score-Based Generative Modeling Through Stochastic Differential Equations》一文中提出可以使用SDE（随机微分方程）来刻画Diffusion model的前向过程，并且用SDE统一了Score-based Model (NCSN)和DDPM的前向过程与反向过程。此外，SDE对应了多个前向过程，即从一张图到某个噪声点的加噪方式有多种，但其中存在一个ODE（常微分方程）形式的前向过程，即不存在随机变量的确定性的前向过程。

本文将总结SDE与DDPM的关系，并给出相应推导

SDE是什么

SDE具体的数学形式如下：
$$dx=f(x,t)dt+g(t)dw\text{\hspace{1em}(1.0)}$$

$f(x,t)$表示自变量$x$随着时间$t$确定性的变化（又被称为drift coefficients），$g(t)$是一项与时间$t$相关的函数（又被称为diffusion coefficients），$dw$为布朗运动的增量，是一个随机项（可以理解为噪声）

SDE与DDPM前向过程的关系

我们将上述部分微分项展开并移位可得

x t + Δ t − x t = f ( x , t ) d t + g ( t ) d w x t + Δ t = x t + f ( x , t ) d t + g ( t ) d w (2.0)

$$\begin{aligned} x_{t+\Delta t}-x_t & = f(x,t)dt+g(t)dw\\ x_{t+\Delta t}& =x_t+f(x,t)dt+g(t)dw\tag{2.0} \end{aligned}$$ xt+Δt​−xt​xt+Δt​​=f(x,t)dt+g(t)dw=xt​+f(x,t)dt+g(t)dw​(2.0)

我们将$x_t$看成是前向过程$t$时刻的图像，则下一时刻$t+\Delta t$的图像$x_{t+\Delta t}$可通过式1.1加噪得到。

接下来，我们将简单推导式2.0与DDPM前向过程的关系，已知DDPM的前向过程为
$$x_{t+\Delta t}=\sqrt{1-{\beta }_{t+\Delta t}}{x}_t+\sqrt{{\beta }_{t+\Delta t}}{ϵ}_t\text{\hspace{1em}(2.1)}$$

设${\overline{\beta }}_{t+\Delta t}=T{\beta }_{t+\Delta t}$，$\Delta t=\frac{1}{T}$，则式2.1为

x t + Δ t = 1 − β t + Δ t x t + β t + Δ t ϵ t = 1 − β ‾ t + Δ t T x t + β t + Δ t ϵ t = 1 − β ‾ t + Δ t Δ t x t + β t + Δ t ϵ t (2.2)

$$\begin{aligned} x_{t+\Delta t}=&\sqrt{1-\beta_{t+\Delta t}} x_{t}+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\\ =&\sqrt{1-\frac{\overline \beta_{t+\Delta t}}{T}} x_{t}+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\\ =&\sqrt{1-\overline \beta_{t+\Delta t}\Delta t} x_{t}+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\tag{2.2} \end{aligned}$$ xt+Δt​===​1−βt+Δt​ ​xt​+βt+Δt​ ​ϵt​1−Tβ​t+Δt​​ ​xt​+βt+Δt​ ​ϵt​1−β​t+Δt​Δt ​xt​+βt+Δt​ ​ϵt​​(2.2)

当$\Delta t$趋近于0，依据等价无穷小代换，式2.2有
x t + Δ t = 1 − β t + Δ t x t + β t + Δ t ϵ t = 1 − β ‾ t + Δ t Δ t x t + β t + Δ t ϵ t ≈ ( 1 − 1 2 β ‾ t + Δ t Δ t ) x t + β t + Δ t ϵ t = x t − 1 2 β ‾ t + Δ t x t d t + β t + Δ t ϵ t (2.3)

$$\begin{aligned} x_{t+\Delta t}=&\sqrt{1-\beta_{t+\Delta t}} x_{t}+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\\ =&\sqrt{1-\overline \beta_{t+\Delta t}\Delta t} x_{t}+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\\ \approx&(1-\frac{1}{2}\overline \beta_{t+\Delta t}\Delta t)x_t+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\\ =&x_t-\frac{1}{2}\overline \beta_{t+\Delta t}x_t dt+\sqrt{\beta_{t+\Delta t}} \epsilon_{t}\tag{2.3} \end{aligned}$$ xt+Δt​==≈=​1−βt+Δt​ ​xt​+βt+Δt​ ​ϵt​1−β​t+Δt​Δt ​xt​+βt+Δt​ ​ϵt​(1−21​β​t+Δt​Δt)xt​+βt+Δt​ ​ϵt​xt​−21​β​t+Δt​xt​dt+βt+Δt​ ​ϵt​​(2.3)

比对式1.4与1.1，则有

f ( x , t ) = − 1 2 β ‾ t + Δ t x t g ( t ) = β t + Δ t d w = ϵ t

$$\begin{aligned} f(x,t)&=-\frac{1}{2}\overline \beta_{t+\Delta t}x_t\\ g(t)&=\sqrt{\beta_{t+\Delta t}}\\ dw&=\epsilon_{t} \end{aligned}$$ f(x,t)g(t)dw​=−21​β​t+Δt​xt​=βt+Δt​ ​=ϵt​​

逆向过程的SDE

前文我们已经介绍了Diffusion model的前向过程可以用SDE描述，本节将推导出逆向过程的SDE形式。

令$dw=\sqrt{\Delta t}ϵ$，由式2.0，可得
$$p(x_{t+\Delta t}|x_t)=\mathcal{N}(x_{t+\Delta t};x_t+f(x_t,\Delta t)\Delta t,g^2(t)\Delta t)\text{\hspace{1em}(3.0)}$$

利用贝叶斯公式，则逆向过程为
q ( x t ∣ x t + Δ t ) = q ( x t + Δ t ∣ x t ) q ( x t ) q ( x t + Δ t ) = q ( x t + Δ t ∣ x t ) exp ⁡ { log ⁡ p ( x t ) − log ⁡ p ( x t + Δ t ) } (3.1)

$$\begin{aligned} q(x_{t}|x_{t+\Delta t})&=\frac{q(x_{t+\Delta t}|x_{t})q(x_{t})}{q(x_{t+\Delta t})}\\ &=q(x_{t+\Delta t}|x_{t})\exp\{\log p(x_t)-\log p(x_{t+\Delta t})\}\tag{3.1} \end{aligned}$$ q(xt​∣xt+Δt​)​=q(xt+Δt​)q(xt+Δt​∣xt​)q(xt​)​=q(xt+Δt​∣xt​)exp{logp(xt​)−logp(xt+Δt​)}​(3.1)

利用泰勒展开，则有

$$\log p(x_{t+\Delta t})\approx \log p(x_t)+(x_{t+\Delta t}-x_t){\nabla }_x\log p(x_t)\text{\hspace{1em}(3.2)}$$

代入式3.1，并且结合式3.0，则有
q ( x t ∣ x t + Δ t ) = q ( x t + Δ t ∣ x t ) exp ⁡ { − ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) } ≈ exp ⁡ { − ( x t + Δ t − x t − f ( x t , t ) Δ t ) 2 + 2 g 2 ( t ) Δ t ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) 2 g 2 ( t ) Δ t } (3.3)

$$\begin{aligned} q(x_{t}|x_{t+\Delta t})&=q(x_{t+\Delta t}|x_{t})\exp \{-(x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)\}\\ &\approx\exp\{-\frac{(x_{t+\Delta t}-x_t-f(x_t,t)\Delta t)^2+2g^2(t)\Delta t (x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)}{2g^2(t)\Delta t }\}\tag{3.3} \end{aligned}$$ q(xt​∣xt+Δt​)​=q(xt+Δt​∣xt​)exp{−(xt+Δt​−xt​)∇x​logp(xt​)}≈exp{−2g2(t)Δt(xt+Δt​−xt​−f(xt​,t)Δt)2+2g2(t)Δt(xt+Δt​−xt​)∇x​logp(xt​)​}​(3.3)

为了后续书写方便，令
a = f ( x t , t ) Δ t b = g 2 ( t ) Δ t

$$\begin{aligned} a&=f(x_t,t)\Delta t\\ b&=g^2(t)\Delta t \end{aligned}$$ ab​=f(xt​,t)Δt=g2(t)Δt​

则有

( x t + Δ t − x t − f ( x t , t ) Δ t ) 2 + 2 g 2 ( t ) Δ t ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) = （ x t + Δ t − x t − a ） 2 + 2 b ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) = ( x t + Δ t − x t ) 2 − 2 a ( x t + Δ t − x t ) + a 2 + 2 b ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) = ( x t + Δ t − x t ) 2 − 2 ( a − b ∇ x log ⁡ p ( x t ) ) ( x t + Δ t − x t ) + ( a − b ) 2 + a 2 − ( a − b ) 2 = ( x t + Δ t − x t − ( a − b ∇ x log ⁡ p ( x t ) ) ) 2 + a 2 − ( a − b ∇ x log ⁡ p ( x t ) ) 2

$$\begin{aligned} &(x_{t+\Delta t}-x_t-f(x_t,t)\Delta t)^2+2g^2(t)\Delta t (x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)\\ &=（x_{t+\Delta t}-x_t-a）^2+2b(x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)\\ &=(x_{t+\Delta t}-x_t)^2-2a(x_{t+\Delta t}-x_t)+a^2+2b(x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)\\ &=(x_{t+\Delta t}-x_t)^2-2(a-b\nabla_{x}\log p(x_t))(x_{t+\Delta t}-x_t)+(a-b)^2+a^2-(a-b)^2\\ &=(x_{t+\Delta t}-x_t-(a-b\nabla_{x}\log p(x_t)))^2+a^2-(a-b\nabla_{x}\log p(x_t))^2 \end{aligned}$$ ​(xt+Δt​−xt​−f(xt​,t)Δt)2+2g2(t)Δt(xt+Δt​−xt​)∇x​logp(xt​)=（xt+Δt​−xt​−a）2+2b(xt+Δt​−xt​)∇x​logp(xt​)=(xt+Δt​−xt​)2−2a(xt+Δt​−xt​)+a2+2b(xt+Δt​−xt​)∇x​logp(xt​)=(xt+Δt​−xt​)2−2(a−b∇x​logp(xt​))(xt+Δt​−xt​)+(a−b)2+a2−(a−b)2=(xt+Δt​−xt​−(a−b∇x​logp(xt​)))2+a2−(a−b∇x​logp(xt​))2​

当$\Delta t$趋近0时，则有

a 2 2 b = f ( x t , t ) 2 Δ t 2 g 2 ( t ) → 0 ( a − b ∇ x log ⁡ p ( x t ) ) 2 2 b = ( f ( x t , t ) − g 2 ( t ) ∇ x log ⁡ p ( x t ) ) Δ t g 2 ( t ) → 0

$$\begin{aligned} \frac{a^2}{2b}&=\frac{f(x_t,t)^2\Delta t}{2g^2(t)} \rightarrow 0\\ \frac{(a-b\nabla_{x}\log p(x_t))^2}{2b}&=\frac{(f(x_t,t)-g^2(t)\nabla_{x}\log p(x_t))\Delta t}{g^2(t)} \rightarrow 0 \end{aligned}$$ 2ba2​2b(a−b∇x​logp(xt​))2​​=2g2(t)f(xt​,t)2Δt​→0=g2(t)(f(xt​,t)−g2(t)∇x​logp(xt​))Δt​→0​

则当$\Delta t$趋近0，式3.3为

q ( x t ∣ x t + Δ t ) = q ( x t + Δ t ∣ x t ) exp ⁡ { − ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) } ≈ exp ⁡ { − ( x t + Δ t − x t − f ( x t , t ) Δ t ) 2 + 2 g 2 ( t ) Δ t ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t ) 2 g 2 ( t ) Δ t } = exp ⁡ { − ( x t + Δ t − x t − ( f ( x t , t ) Δ t − g 2 ( t ) Δ t ∇ x log ⁡ p ( x t ) ) ) 2 2 g 2 ( t ) Δ t } (3.4)

$$\begin{aligned} q(x_{t}|x_{t+\Delta t})&=q(x_{t+\Delta t}|x_{t})\exp \{-(x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)\}\\ &\approx\exp\{-\frac{(x_{t+\Delta t}-x_t-f(x_t,t)\Delta t)^2+2g^2(t)\Delta t (x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_t)}{2g^2(t)\Delta t }\}\\ &=\exp\{-\frac{(x_{t+\Delta t}-x_t-(f(x_t,t)\Delta t-g^2(t)\Delta t\nabla_{x}\log p(x_t)))^2}{2g^2(t)\Delta t}\}\tag{3.4} \end{aligned}$$ q(xt​∣xt+Δt​)​=q(xt+Δt​∣xt​)exp{−(xt+Δt​−xt​)∇x​logp(xt​)}≈exp{−2g2(t)Δt(xt+Δt​−xt​−f(xt​,t)Δt)2+2g2(t)Δt(xt+Δt​−xt​)∇x​logp(xt​)​}=exp{−2g2(t)Δt(xt+Δt​−xt​−(f(xt​,t)Δt−g2(t)Δt∇x​logp(xt​)))2​}​(3.4)

则有

$$q(x_t|x_{t+\Delta t})=\mathcal{N}(x_t|x_{t+\Delta t}-f(x_t,t)\Delta t+g^2(t)\Delta t{\nabla }_x\log p(x_t),2g^2(t)\Delta t)\text{\hspace{1em}(3.5)}$$

设噪声$z$服从标准正态分布，则式3.5写成SDE的形式为

x t = x t + Δ t − f ( x t , t ) Δ t + g 2 ( t ) Δ t ∇ x log ⁡ p ( x t ) + g ( t ) 2 Δ t z d x = ( f ( x t , t ) − g 2 ( t ) ∇ x log ⁡ p ( x t ) ) d t − g ( t ) 2 Δ t z = ( f ( x t , t ) − g 2 ( t ) ∇ x log ⁡ p ( x t ) ) d t + g ( t ) d w ‾

$$\begin{aligned} x_t&=x_{t+\Delta t}-f(x_t,t)\Delta t+g^2(t)\Delta t\nabla_{x}\log p(x_t)+g(t)\sqrt{2\Delta t}z\\ dx&=(f(x_t,t)-g^2(t)\nabla_{x}\log p(x_t))dt-g(t)\sqrt{2\Delta t}z\\ &=(f(x_t,t)-g^2(t)\nabla_{x}\log p(x_t))dt+g(t)d\overline{w} \end{aligned}$$ xt​dx​=xt+Δt​−f(xt​,t)Δt+g2(t)Δt∇x​logp(xt​)+g(t)2Δt ​z=(f(xt​,t)−g2(t)∇x​logp(xt​))dt−g(t)2Δt ​z=(f(xt​,t)−g2(t)∇x​logp(xt​))dt+g(t)dw​

${\nabla }_{x_t}\log p(x_t)$与DDPM预测的噪声$ϵ$的关系

score base model一般会用神经网络拟合${\nabla }_{x_t}\log p(x_t)$，DDPM其实是一种特殊的score base model，已知DDPM的前向过程为

$$x_t=\sqrt{{\bar{\alpha }}_t}{x}_0+\sqrt{1-{\bar{\alpha }}_t}{ϵ}_t$$

依据Tweedie方法，我们有

α ˉ t x 0 = x t + ( 1 − α ˉ t ) ∇ x log ⁡ p ( x t )

$$\begin{aligned} \sqrt{\bar \alpha_t}x_0=x_t+(1-\bar\alpha_t)\nabla_{x}\log p(x_t) \end{aligned}$$ αˉt​ ​x0​=xt​+(1−αˉt​)∇x​logp(xt​)​
进而有
$$x_t=\sqrt{{\bar{\alpha }}_t}{x}_0-(1-{\bar{\alpha }}_t){\nabla }_x\log p(x_t)\text{\hspace{1em}(4.0)}$$

结合式1.0与1.1，则有

$${\nabla }_{x_t}\log p(x_t)=-\frac{1}{\sqrt{1-{\bar{\alpha }}_t}}{ϵ}_t\text{\hspace{1em}(4.1)}$$

逆向过程SDE与DDPM逆向过程的关系

在进行正式的推导前，我们先对式3.1做个简单的变化，利用泰勒展开，则有
$$\log p(x_t)\approx \log p(x_{t+\Delta t})+(x_t-x_{t+\Delta t}){\nabla }_x\log p(x_{t+\Delta t})\text{\hspace{1em}(5.0)}$$

代入式3.1，并结合式3.0，则有
q ( x t ∣ x t + Δ t ) = = q ( x t + Δ t ∣ x t ) exp ⁡ { log ⁡ p ( x t ) − log ⁡ p ( x t + Δ t ) = q ( x t + Δ t ∣ x t ) exp ⁡ { − ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t + Δ t ) } ≈ exp ⁡ { − ( x t + Δ t − x t − f ( x t , t ) Δ t ) 2 + 2 g 2 ( t ) Δ t ( x t + Δ t − x t ) ∇ x log ⁡ p ( x t + Δ t ) 2 g 2 ( t ) Δ t } (5.1)

$$\begin{aligned} q(x_{t}|x_{t+\Delta t})&==q(x_{t+\Delta t}|x_{t})\exp\{\log p(x_t)-\log p(x_{t+\Delta t})\\ &=q(x_{t+\Delta t}|x_{t})\exp \{-(x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_{t+\Delta t})\}\\ &\approx\exp\{-\frac{(x_{t+\Delta t}-x_t-f(x_t,t)\Delta t)^2+2g^2(t)\Delta t (x_{t+\Delta t}-x_t)\nabla_{x}\log p(x_{t+\Delta t})}{2g^2(t)\Delta t }\}\tag{5.1} \end{aligned}$$ q(xt​∣xt+Δt​)​==q(xt+Δt​∣xt​)exp{logp(xt​)−logp(xt+Δt​)=q(xt+Δt​∣xt​)exp{−(xt+Δt​−xt​)∇x​logp(xt+Δt​)}≈exp{−2g2(t)Δt(xt+Δt​−xt​−f(xt​,t)Δt)2+2g2(t)Δt(xt+Δt​−xt​)∇x​logp(xt+Δt​)​}​(5.1)

因此我们可将式3.5重写为
$$q(x_t|x_{t+\Delta t})=\mathcal{N}(x_t|x_{t+\Delta t}-f(x_t,t)\Delta t+g^2(t)\Delta t{\nabla }_x\log p(x_{t+\Delta t}),2g^2(t)\Delta t)\text{\hspace{1em}(5.2)}$$

已知用SDE表示DDPM的前向过程时，有

f ( x , t ) = − 1 2 β ‾ t + Δ t x t g ( t ) = β t + Δ t

$$\begin{aligned} f(x,t)&=-\frac{1}{2}\overline \beta_{t+\Delta t}x_t\\ g(t)&=\sqrt{\beta_{t+\Delta t}} \end{aligned}$$ f(x,t)g(t)​=−21​β​t+Δt​xt​=βt+Δt​ ​​

其中
$${\overline{\beta }}_{t+\Delta t}=T{\beta }_{t+\Delta t}=\frac{{\beta }_{t+\Delta t}}{\Delta t}$$

$T=\frac{1}{\Delta t}$，设$z$服从标准正态分布，代入式5.2并结合式4.1，当$\Delta t$趋近于0时，有

x t = x t + Δ t − f ( x t , t ) Δ t + g 2 ( t ) Δ t ∇ x log ⁡ p ( x t + Δ t ) + g ( t ) 2 Δ t z x t = x t + Δ t + 1 2 β ‾ t + Δ t x t Δ t + β t + Δ t Δ t ∇ x log ⁡ p ( x t + Δ t ) + 2 β t + Δ t Δ t z ( 1 − 1 2 β ‾ t + Δ t Δ t ) x t = x t + Δ t + β t + Δ t Δ t ∇ x log ⁡ p ( x t + Δ t ) + 2 β t + Δ t Δ t z 1 − β t + Δ t x t ≈ x t + Δ t + β t + Δ t Δ t ∇ x log ⁡ p ( x t + Δ t ) + 2 β t + Δ t Δ t z x t ≈ 1 1 − β t + Δ t ( x t + Δ t + β t + Δ t Δ t ∇ x log ⁡ p ( x t + Δ t ) ) + 2 β t + Δ t Δ t 1 − β t + Δ t z x t ≈ 1 1 − β t + Δ t ( x t + Δ t + β t + Δ t Δ t ∇ x log ⁡ p ( x t + Δ t ) ) + 2 β t + Δ t Δ t 1 − β t + Δ t z x t ≈ 1 1 − β t + Δ t ( x t + Δ t − β t + Δ t Δ t 1 − α ˉ t ϵ t + Δ t ) ) + 2 β t + Δ t Δ t 1 − β t + Δ t z

$$\begin{aligned} x_t&=x_{t+\Delta t}-f(x_t,t)\Delta t+g^2(t)\Delta t\nabla_{x}\log p(x_{t+\Delta t})+g(t)\sqrt{2\Delta t}z\\ x_t&=x_{t+\Delta t}+\frac{1}{2}\overline \beta_{t+\Delta t}x_t\Delta t+\beta_{t+\Delta t}\Delta t \nabla_{x}\log p(x_{t+\Delta t})+\sqrt{2\beta_{t+\Delta t}\Delta t}z\\ (1-\frac{1}{2}\overline \beta_{t+\Delta t}\Delta t)x_t&=x_{t+\Delta t}+\beta_{t+\Delta t}\Delta t \nabla_{x}\log p(x_{t+\Delta t})+\sqrt{2\beta_{t+\Delta t}\Delta t}z\\ \sqrt{1- \beta_{t+\Delta t}}x_t&\approx x_{t+\Delta t}+\beta_{t+\Delta t}\Delta t \nabla_{x}\log p(x_{t+\Delta t})+\sqrt{2\beta_{t+\Delta t}\Delta t}z\\ x_t&\approx \frac{1}{\sqrt{1- \beta_{t+\Delta t}}}(x_{t+\Delta t}+\beta_{t+\Delta t}\Delta t \nabla_{x}\log p(x_{t+\Delta t}))+\sqrt{\frac{2\beta_{t+\Delta t}\Delta t}{1-\beta_{t+\Delta t}}}z\\ x_t&\approx \frac{1}{\sqrt{1- \beta_{t+\Delta t}}}(x_{t+\Delta t}+\beta_{t+\Delta t}\Delta t \nabla_{x}\log p(x_{t+\Delta t}))+\sqrt{\frac{2\beta_{t+\Delta t}\Delta t}{1-\beta_{t+\Delta t}}}z\\ x_t &\approx \frac{1}{\sqrt{1- \beta_{t+\Delta t}}}(x_{t+\Delta t} -\frac{\beta_{t+\Delta t}\Delta t}{\sqrt{1-\bar\alpha_t}}\epsilon_{t+\Delta t}))+\sqrt{\frac{2\beta_{t+\Delta t}\Delta t}{1-\beta_{t+\Delta t}}}z \end{aligned}$$ xt​xt​(1−21​β​t+Δt​Δt)xt​1−βt+Δt​ ​xt​xt​xt​xt​​=xt+Δt​−f(xt​,t)Δt+g2(t)Δt∇x​logp(xt+Δt​)+g(t)2Δt ​z=xt+Δt​+21​β​t+Δt​xt​Δt+βt+Δt​Δt∇x​logp(xt+Δt​)+2βt+Δt​Δt ​z=xt+Δt​+βt+Δt​Δt∇x​logp(xt+Δt​)+2βt+Δt​Δt ​z≈xt+Δt​+βt+Δt​Δt∇x​logp(xt+Δt​)+2βt+Δt​Δt ​z≈1−βt+Δt​ ​1​(xt+Δt​+βt+Δt​Δt∇x​logp(xt+Δt​))+1−βt+Δt​2βt+Δt​Δt​ ​z≈1−βt+Δt​ ​1​(xt+Δt​+βt+Δt​Δt∇x​logp(xt+Δt​))+1−βt+Δt​2βt+Δt​Δt​ ​z≈1−βt+Δt​ ​1​(xt+Δt​−1−αˉt​ ​βt+Δt​Δt​ϵt+Δt​))+1−βt+Δt​2βt+Δt​Δt​ ​z​

当$\Delta t=1$时，则有

$$x_t\approx \frac{1}{\sqrt{1-{\beta }_{t+1}}}(x_{t+1}-\frac{{\beta }_{t+1}}{\sqrt{1-{\bar{\alpha }}_t}}{ϵ}_{t+1}))+\sqrt{\frac{2{\beta }_{t+1}}{1-{\beta }_{t+1}}}z$$

不过这种约等于号是真的很膈应，是不能做完全等价的。

Probability Flow (PF) ODE

有多种SDE，可以将一张图像变为某个噪声点，其中也包括一个ODE（即去除掉SDE中布朗运动增量）

对于前向过程

$$dx=f(x,t)dt+g(t)dw$$

由Fokker-Planck方程可得

∂ p ( x , t ) ∂ t = − ∑ i ∂ ∂ x i [ f i ( x , t ) p ( x , t ) ] + 1 2 ∑ i , j ∂ 2 ∂ x i x j { [ g 2 ( t ) I ] i j p ( x , t ) } = − ∑ i ∂ ∂ x i [ f i ( x , t ) p ( x , t ) ] + 1 2 ∑ i ∂ 2 ∂ x i 2 [ g 2 ( t ) p ( x , t ) ]

$$\begin{aligned} \frac{\partial p(x,t)}{\partial t} &= -\sum_{i} \frac{\partial}{\partial x_i}[f_i(x,t)p(x,t)] + \frac{1}{2} \sum_{i,j}\frac{\partial^2}{\partial x_i x_j}\left\{[g^2(t)I]_{ij} p(x,t)\right\} \\ &= -\sum_{i} \frac{\partial}{\partial x_i}[f_i(x,t)p(x,t)] + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[g^2(t) p(x,t)] \end{aligned}$$ ∂t∂p(x,t)​​=−i∑​∂xi​∂​[fi​(x,t)p(x,t)]+21​i,j∑​∂xi​xj​∂2​{[g2(t)I]ij​p(x,t)}=−i∑​∂xi​∂​[fi​(x,t)p(x,t)]+21​i∑​∂xi2​∂2​[g2(t)p(x,t)]​

对上述式子做个等价变换，则有
∂ p ( x , t ) ∂ t = − ∑ i ∂ ∂ x i [ f i ( x , t ) p ( x , t ) ] + 1 2 ∑ i ∂ 2 ∂ x i 2 [ g 2 ( t ) p ( x , t ) ] = − ∑ i ∂ ∂ x i [ f i ( x , t ) p ( x , t ) ] + 1 2 ∑ i ∂ 2 ∂ x i 2 [ ( g 2 ( t ) − σ 2 ( t ) ) p ( x , t ) ] + 1 2 ∑ i ∂ 2 ∂ x i 2 [ σ 2 ( t ) p ( x , t ) ] = − ∑ i ∂ ∂ x i [ f i ( x , t ) p ( x , t ) ] + 1 2 ∑ i ∂ ∂ x i ( g 2 ( t ) − σ 2 ( t ) ) ∂ ∂ x i p ( x , t ) + 1 2 ∑ i ∂ 2 ∂ x i 2 [ σ 2 ( t ) p ( x , t ) ] = − ∑ i ∂ ∂ x i [ f i ( x , t ) p ( x , t ) ] + 1 2 ∑ i ∂ ∂ x i ( g 2 ( t ) − σ 2 ( t ) ) p ( x , t ) ∂ ∂ x i log ⁡ p ( x , t ) + 1 2 ∑ i ∂ 2 ∂ x i 2 [ σ 2 ( t ) p ( x , t ) ] = − ∑ i ∂ ∂ x i [ ( f i ( x , t ) − 1 2 ( g 2 ( t ) − σ 2 ( t ) ) ∂ ∂ x i log ⁡ p ( x , t ) ) p ( x , t ) ] + 1 2 ∑ i ∂ 2 ∂ x i 2 [ σ 2 ( t ) p ( x , t ) ]

$$\begin{aligned} \frac{\partial p(x,t)}{\partial t} &= -\sum_{i} \frac{\partial}{\partial x_i}[f_i(x,t)p(x,t)] + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[g^2(t) p(x,t)] \\ &= -\sum_{i} \frac{\partial}{\partial x_i}[f_i(x,t)p(x,t)] + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[(g^2(t) - \sigma^2(t)) p(x,t)] + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[\sigma^2(t) p(x,t)] \\ &= -\sum_{i} \frac{\partial}{\partial x_i}[f_i(x,t)p(x,t)] + \frac{1}{2} \sum_{i}\frac{\partial}{\partial x_i} (g^2(t) - \sigma^2(t)) \frac{\partial}{\partial x_i} p(x,t) + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[\sigma^2(t) p(x,t)] \\ &= -\sum_{i} \frac{\partial}{\partial x_i}[f_i(x,t)p(x,t)] + \frac{1}{2} \sum_{i}\frac{\partial}{\partial x_i}(g^2(t) - \sigma^2(t)) p(x,t) \frac{\partial}{\partial x_i} \log p(x,t) + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[\sigma^2(t) p(x,t)] \\ &= -\sum_{i} \frac{\partial}{\partial x_i} \left[ \left(f_i(x,t) - \frac{1}{2}(g^2(t) - \sigma^2(t)) \frac{\partial}{\partial x_i} \log p(x,t) \right)p(x,t) \right] + \frac{1}{2} \sum_{i}\frac{\partial^2}{\partial x_i^2}[\sigma^2(t) p(x,t)] \end{aligned}$$ ∂t∂p(x,t)​​=−i∑​∂xi​∂​[fi​(x,t)p(x,t)]+21​i∑​∂xi2​∂2​[g2(t)p(x,t)]=−i∑​∂xi​∂​[fi​(x,t)p(x,t)]+21​i∑​∂xi2​∂2​[(g2(t)−σ2(t))p(x,t)]+21​i∑​∂xi2​∂2​[σ2(t)p(x,t)]=−i∑​∂xi​∂​[fi​(x,t)p(x,t)]+21​i∑​∂xi​∂​(g2(t)−σ2(t))∂xi​∂​p(x,t)+21​i∑​∂xi2​∂2​[σ2(t)p(x,t)]=−i∑​∂xi​∂​[fi​(x,t)p(x,t)]+21​i∑​∂xi​∂​(g2(t)−σ2(t))p(x,t)∂xi​∂​logp(x,t)+21​i∑​∂xi2​∂2​[σ2(t)p(x,t)]=−i∑​∂xi​∂​[(fi​(x,t)−21​(g2(t)−σ2(t))∂xi​∂​logp(x,t))p(x,t)]+21​i∑​∂xi2​∂2​[σ2(t)p(x,t)]​

利用Fokker-Planck方程的对应关系，则有
$$dx=\left(f(x,t)-\frac{1}{2}(g^2(t)-{\sigma }^2(t)){\nabla }_x\log p(x,t)\right)dt+\sigma (t)dw$$

特别的，当$\delta (t)=0$时，则有
$$dx=\left(f(x,t)-\frac{1}{2}{g}^2(t){\nabla }_x\log p(x,t)\right)dt$$
上述式子又被称为Probability Flow (PF) ODE