扩散去噪：从随机噪声中逆向重建的随机微分方程

Token炼金师2026-06-24 9:46

摘要

扩散去噪不是"逐步擦除噪声"的直觉过程，而是求解随机微分方程（SDE）的逆向过程------前向过程按噪声调度表 αˉt\bar\alpha_tαˉt 逐步加噪，逆向过程用训练好的 ϵθ\epsilon_\thetaϵθ 估计每步噪声再从 xtx_txt 中扣除，数学本质是Langevin动力学与分数匹配的统一。本文从前向加噪的马尔可夫链、逆向SDE的推导、噪声预测与分数函数的等价性、信噪比与噪声调度的关系、去噪一步的几何意义、VE/VP/VPSDE三种扩散框架，到ComfyUI中的去噪循环实现，逐层拆解扩散去噪的底层原理。

1. 前向过程：马尔可夫链与噪声调度

前向过程定义了如何将干净图像 x0x_0x0 逐步加噪到纯噪声 xTx_TxT。每一步是一个条件高斯转移：

q(xt∣xt−1)=N(xt;1−βtxt−1,βtI)q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{1-\beta_t} x_{t-1}, \beta_t I)q(xt∣xt−1)=N(xt;1−βt xt−1,βtI)

通过重参数化可一步到位：xt=αˉtx0+1−αˉtϵx_t = \sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t} \epsilonxt=αˉt x0+1−αˉt ϵ，其中 αˉt=∏s=1t(1−βs)\bar\alpha_t = \prod_{s=1}^{t}(1-\beta_s)αˉt=∏s=1t(1−βs)。
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.default>*{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-8Eg8lNubpkkGMKtQ .default span{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-8Eg8lNubpkkGMKtQ .default tspan{fill:#000000!important;} x_0 干净图像
x_1 + noise
x_2 + noise
...
x_T 纯噪声
任意步直达
x_t = sqrt alpha_bar_t * x_0 + sqrt 1 - alpha_bar_t * epsilon

// 来源：Ho et al. "Denoising Diffusion Probabilistic Models" (2020) Section 2

python 复制代码 
# 前向过程: 逐步加噪与一步直达
import torch

def forward_process(x_0, beta_schedule, t):
    """一步从x_0到x_t"""
    # beta_schedule: [T] 每步的噪声方差
    alpha = 1.0 - beta_schedule                    # alpha_t = 1 - beta_t
    alpha_bar = torch.cumprod(alpha, dim=0)         # alpha_bar_t = prod(1-beta_s)
    alpha_bar_t = alpha_bar[t]
    # 一步直达: x_t = sqrt(alpha_bar_t) * x_0 + sqrt(1 - alpha_bar_t) * eps
    eps = torch.randn_like(x_0)
    x_t = torch.sqrt(alpha_bar_t) * x_0 + torch.sqrt(1 - alpha_bar_t) * eps
    return x_t, eps  # eps是用于训练的目标

# SD1.5的噪声调度 (linear, T=1000)
# beta_1 = 0.00085, beta_1000 = 0.012
# alpha_bar_1 ≈ 0.999, alpha_bar_1000 ≈ 0.007
# t=1时几乎无噪声, t=1000时几乎纯噪声

2. 逆向过程的SDE推导

逆向去噪是前向过程的概率逆，可以用SDE严格推导。前向SDE为：

dx=f(x,t)dt+g(t)dwdx = f(x,t)dt + g(t)d\mathbf{w}dx=f(x,t)dt+g(t)dw

Anderson(1982)定理给出逆向SDE：

dx=f(x,t)−g(t)2∇xlog⁡pt(x)dt+g(t)dwˉdx = f(x,t) - g(t)\^2 \\nabla_x \\log p_t(x)dt + g(t)d\bar{\mathbf{w}}dx=f(x,t)−g(t)2∇xlogpt(x)dt+g(t)dwˉ
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.default>*{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-JgKOoNEXXhbhAekM .default span{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-JgKOoNEXXhbhAekM .default tspan{fill:#000000!important;} 前向SDE
dx = f x t dt + g t dw
逆向SDE Anderson 1982
dx = f - g^2 grad log p_t dt + g d w_bar
核心: 需要分数函数 grad_x log p_t x
分数函数不可直接计算
但等价于去噪!
score = -epsilon / sqrt 1 - alpha_bar_t
训练epsilon_theta估计噪声 = 估计分数函数

// 来源：Song et al. "Score-Based Generative Modeling through SDEs" (2021)

python 复制代码 
# 分数函数与噪声预测的等价性
# x_t = sqrt(alpha_bar) * x_0 + sqrt(1 - alpha_bar) * eps
# p(x_t)的分数函数:
#   score(x_t, t) = grad_{x_t} log p(x_t)
#                 = -eps / sqrt(1 - alpha_bar_t)
#   (高斯分布的分数函数有解析形式)
#
# 因此: 训练 epsilon_theta(x_t, t) ≈ eps
#   等价于训练 score_theta(x_t, t) ≈ score(x_t, t)
#
# 去噪 = 逆向SDE求解 = 用分数函数引导随机游走回到数据分布

def score_from_epsilon(eps_pred, alpha_bar_t):
    """从噪声预测转换为分数函数"""
    return -eps_pred / torch.sqrt(1 - alpha_bar_t)

3. DDPM去噪：离散马尔可夫转移

DDPM将逆向SDE离散化为有限步马尔可夫转移，每一步从 xtx_txt 估计 xt−1x_{t-1}xt−1：

xt−1=1αt(xt−1−αt1−αˉtϵθ(xt,t))+σtzx_{t-1} = \frac{1}{\sqrt{\alpha_t}} \left( x_t - \frac{1-\alpha_t}{\sqrt{1-\bar\alpha_t}} \epsilon_\theta(x_t, t) \right) + \sigma_t zxt−1=αt 1(xt−1−αˉt 1−αtϵθ(xt,t))+σtz
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eps_theta x_t t 预测噪声
x_t - 1-alpha / sqrt 1-alpha_bar * eps_pred
/ sqrt alpha_t

  • sigma_t * z 随机性注入
    x_t-1
    sigma_t
    方差选择
    小: sigma = beta 后验方差
    大: sigma = beta_tilde 最大方差

// 来源：Ho et al. (2020) Algorithm 2

python 复制代码 
# DDPM单步去噪
def ddpm_step(x_t, t, eps_pred, beta_t, alpha_t, alpha_bar_t):
    """DDPM去噪一步: x_t -> x_{t-1}"""
    # 确定性部分: 从x_t中扣除预测噪声
    coef1 = 1.0 / torch.sqrt(alpha_t)
    coef2 = (1 - alpha_t) / torch.sqrt(1 - alpha_bar_t)
    x_pred = coef1 * (x_t - coef2 * eps_pred)
    # 随机性部分: 注入小量噪声
    sigma_t = torch.sqrt(beta_t)  # 后验标准差
    z = torch.randn_like(x_t) if t > 0 else 0  # 最后一步不加噪声
    return x_pred + sigma_t * z

# 关键: DDPM每步注入随机噪声
# 这是随机采样器, 不是确定性的
# 随机性来自Langevin动力学

4. DDIM去噪：确定性逆向的数学本质

DDIM将逆向过程中的随机项设为零，得到确定性映射。等价于求解逆向常微分方程（ODE）。

xt−1=αˉt−1xt−1−αˉtϵθαˉt⏟预测的x0+1−αˉt−1ϵθx_{t-1} = \sqrt{\bar\alpha_{t-1}} \underbrace{\frac{x_t - \sqrt{1-\bar\alpha_t}\epsilon_\theta}{\sqrt{\bar\alpha_t}}}{\text{预测的}x_0} + \sqrt{1-\bar\alpha{t-1}} \epsilon_\thetaxt−1=αˉt−1 预测的x0 αˉt xt−1−αˉt ϵθ+1−αˉt−1 ϵθ
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.default>*{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-mWN5MTPbRIGgGGr7 .default span{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-mWN5MTPbRIGgGGr7 .default tspan{fill:#000000!important;} DDIM
逆向ODE求解
确定性: 相同输入永远相同输出
eta参数控制随机性
eta=0: 纯ODE 确定性
eta=1: 等价DDPM 随机性
子步采样
1000步可跳至50步
跳步不破坏一致性
ODE轨迹不依赖步数

步数只影响精度

// 来源：Song et al. "Denoising Diffusion Implicit Models" (2021)

python 复制代码 
# DDIM单步去噪
def ddim_step(x_t, t, t_prev, eps_pred, alpha_bar_t, alpha_bar_prev, eta=0.0):
    """DDIM去噪一步: x_t -> x_{t_prev}"""
    # 预测x_0 (从x_t和预测噪声反推)
    x0_pred = (x_t - torch.sqrt(1 - alpha_bar_t) * eps_pred) / torch.sqrt(alpha_bar_t)
    # 方差项
    sigma = eta * torch.sqrt(
        (1 - alpha_bar_prev) / (1 - alpha_bar_t) * (1 - alpha_bar_t / alpha_bar_prev)
    )
    # 确定性部分 + 可选随机性
    dir_xt = torch.sqrt(1 - alpha_bar_prev - sigma**2) * eps_pred
    noise = sigma * torch.randn_like(x_t) if sigma > 0 else 0
    return torch.sqrt(alpha_bar_prev) * x0_pred + dir_xt + noise

# eta=0: 纯确定性ODE, 可复现
# eta=1: 等价DDPM, 随机采样
# ComfyUI的eta参数就是这个

5. 信噪比与噪声调度的深层关系

αˉt\bar\alpha_tαˉt 定义了时刻t的信噪比：SNR(t)=αˉt/(1−αˉt)\text{SNR}(t) = \bar\alpha_t / (1 - \bar\alpha_t)SNR(t)=αˉt/(1−αˉt)。噪声调度的选择直接影响去噪轨迹的质量。
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.default>*{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-X3cVbsIhsvnZGljK .default span{fill:#faf9f5!important;stroke:#ffffff!important;color:#000000!important;stroke-width:0px!important;}#mermaid-svg-X3cVbsIhsvnZGljK .default tspan{fill:#000000!important;} 信噪比 SNR = alpha_bar / 1 - alpha_bar
噪声调度策略
Linear: beta均匀增长
Cosine: alpha_bar = cos^2调度
早期SNR过高 细节丢失快
SNR平滑下降 细节保留好
截断调度
SD1.5: beta_1=0.00085 beta_T=0.012

避免t=0时beta=0导致数值问题

// 来源：Nichol & Dhariwal "Improved DDPM" (2021) + SD训练配置

python 复制代码 
# 不同噪声调度的SNR曲线对比
def linear_schedule(T=1000, beta_start=0.00085, beta_end=0.012):
    betas = torch.linspace(beta_start, beta_end, T)
    alpha_bar = torch.cumprod(1 - betas, dim=0)
    return alpha_bar

def cosine_schedule(T=1000, s=0.008):
    steps = torch.linspace(0, T, T + 1)
    alpha_bar = torch.cos((steps / T + s) / (1 + s) * math.pi / 2) ** 2
    alpha_bar = alpha_bar / alpha_bar[0]
    return alpha_bar[1:]

# SNR(t=500) 对比:
# Linear: SNR ≈ 0.18 (信噪比极低, 半程已接近纯噪声)
# Cosine: SNR ≈ 0.85 (信噪比适中, 细节保留更好)
# Cosine调度在ImageNet上FID提升约15%
# SD1.5使用的是scaled linear (接近cosine的效果)

6. 去噪一步的几何意义：梯度上升与流形投影

单步去噪的几何意义：ϵθ\epsilon_\thetaϵθ 估计的噪声方向指向数据流形，去噪一步等价于沿分数函数方向走一步梯度上升。
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几何意义
epsilon_theta指向数据流形法线方向
扣除噪声 = 沿流形方向移动
注入小噪声 = Langevin探索
类比: 带噪声的梯度下降
梯度下降: 确定性沿梯度走
Langevin: 梯度 + 随机扰动 = 避免局部极小
为什么需要随机性?
确定性ODE可能陷入流形局部
随机扰动帮助探索流形邻域
多样性来源: 不同的噪声实现->不同结果

// 来源：Song & Ermon (2019) + Welling & Teh (2011)

python 复制代码 
# 去噪的Langevin动力学解释
def langevin_step(x_t, score_fn, step_size, noise_scale):
    """Langevin动力学一步 = 梯度上升 + 随机扰动"""
    # 确定性: 沿分数函数方向走一步
    grad = score_fn(x_t)  # = -eps / sqrt(1-alpha_bar)
    x_determ = x_t + 0.5 * step_size * grad
    # 随机性: 注入噪声探索邻域
    x_next = x_determ + noise_scale * torch.randn_like(x_t)
    return x_next

# DDPM = 离散Langevin动力学 (每步注入噪声)
# DDIM = 确定性梯度上升 (不注入噪声)
# 两者统一于逆向SDE/ODE框架

7. 三种扩散框架：VE/VP/VPSDE

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Variance Exploding SDE
Variance Preserving SDE
sub-VP SDE
前向: dx = sqrt d sigma^2 dt * dw

方差爆炸 均值不变
前向: dx = -0.5 beta_t x dt + sqrt beta_t dw

方差保持 均值衰减
前向: VP的变体 方差有界但更小
Score SDE NCSN
DDPM SD1.5 SDXL
很少单独使用

// 来源：Song et al. (2021) Table 1

python 复制代码 
# VE vs VP的数学对比
# VE (Variance Exploding):
#   dx = sqrt(d(sigma^2)/dt) * dw
#   x_t = x_0 + sigma_t * eps  (均值不变, 方差爆炸)
#   逆向: 需要估计score(x_t, sigma_t)
#   典型: NCSN, Score SDE

# VP (Variance Preserving):
#   dx = -0.5 * beta_t * x * dt + sqrt(beta_t) * dw
#   x_t = sqrt(alpha_bar_t) * x_0 + sqrt(1-alpha_bar_t) * eps
#   方差保持: Var(x_t) = alpha_bar_t + (1-alpha_bar_t) = 1
#   典型: DDPM, SD1.5, SDXL

# SD选择VP的原因:
# 1. 方差有界, 训练更稳定
# 2. 与VAE潜空间配合(潜表示方差已归一化)
# 3. 噪声调度alpha_bar_t直观控制SNR

8. ComfyUI去噪循环：从调度到采样器的完整路径

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Scheduler: 计算timesteps
去噪循环
UNet: eps_theta x_t t c
CFG: eps_u + w * eps_c - eps_u
Sampler: DDPM or DDIM Step
x_t-1
x_0_hat
VAE Decode

// 来源：ComfyUI / comfy/samplers.py + comfy/k_diffusion/sampling.py

python 复制代码 
# ComfyUI去噪循环的简化实现
def denoise_loop(model, scheduler, x_T, cond, uncond, cfg_scale, steps):
    """完整的去噪循环"""
    timesteps = scheduler.get_timesteps(steps)  # 噪声调度
    x_t = x_T
    for t in timesteps:
        # 1. UNet预测噪声 (CFG: 两次前向)
        eps_c = model(x_t, t, **cond)
        eps_u = model(x_t, t, **uncond)
        eps_cfg = eps_u + cfg_scale * (eps_c - eps_u)
        # 2. 采样器计算x_{t-1}
        x_t = scheduler.step(eps_cfg, t, x_t)
    return x_t  # x_0_hat

# scheduler.step的选择:
# DDPM: 随机性, 每步注入噪声, 需1000步
# DDIM: 确定性, eta=0, 可20步完成
# DPM++: 高阶ODE求解器, 10-20步高质量
# Euler: 一阶ODE, 最简单最快

总结

扩散去噪的底层原理：前向过程是按噪声调度 αˉt\bar\alpha_tαˉt 加噪的马尔可夫链（定义）→ 逆向过程是Anderson定理给出的逆向SDE（推导）→ 分数函数与噪声预测等价 score=−ϵ/1−αˉt\text{score} = -\epsilon/\sqrt{1-\bar\alpha_t}score=−ϵ/1−αˉt （桥梁）→ DDPM是离散Langevin动力学（随机采样）→ DDIM是逆向ODE（确定性映射）→ 噪声调度控制SNR曲线形状（调度）→ VP框架保持方差有界（SD的选择）。核心洞察：去噪不是"擦除噪声"，而是沿分数函数方向做Langevin随机游走回到数据流形------随机性不是缺陷，而是探索流形邻域的必要机制。

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