前言

生成式建模的扩散思想实际上已经在2015年（Sohl-Dickstein等人）提出，然而，直到2019年斯坦福大学（Song等人）、2020年Google Brain（Ho等人）才改进了这个方法，从此引发了生成式模型的新潮流。目前，包括OpenAI的GLIDE和DALL-E 2，海德堡大学的Latent Diffusion和Google Brain的ImageGen，都基于diffusion模型，并可以得到高质量的生成效果。本文以下讲解主要基于DDPM，并适当地增加一些目前有效的改进内容。

基本原理

扩散模型包括两个步骤：

固定的（或预设的）前向扩散过程q：该过程会逐渐将高斯噪声添加到图像中，直到最终得到纯噪声。

可训练的反向去噪扩散过程：训练一个神经网络，从纯噪音开始逐渐去噪，直到得到一个真实图像。

前向与后向的步数由下标 t定义，并且有预先定义好的总步数 T（DDPM原文中为1000）。

t=0 时为从数据集中采样得到的一张真实图片， t=T 时近似为一张纯粹的噪声。

2.1 直观理解

为了看懂扩散模型查了很多资料，但是要么就是大量的数学公式，一行行公式推完了还是不知道它想干啥。要么就是高视角，上来就和能量模型，VAE放一块儿对比说共同点和不同点，看完还是云里雾里。然而事实上下面几句话就能把扩散模型说明白了
扩散模型的目的是什么？
学习从纯噪声生成图片的方法
扩散模型是怎么做的？
训练一个U-Net，接受一系列加了噪声的图片，学习预测所加的噪声
前向过程在干啥？
逐步向真实图片添加噪声最终得到一个纯噪声
对于训练集中的每张图片，都能生成一系列的噪声程度不同的加噪图片
在训练时，这些 【不同程度的噪声图片 + 生成它们所用的噪声】 是实际的训练样本
反向过程在干啥？
训练好模型后，采样、生成图片

2.2 数学形式

2.2.1 前向过程

是真实数据分布（也就是真实的大量图片），从这个分布中采样即可得到一张真实图片 。我们定义前向扩散过程为 ，即每一个step向图片添加噪声的过程，并定义好一系列，则有：

其中，N为正态分布，均值和方差分别为，因此通过采样标准正态分布，有：

2.2.2 反向过程

那么问题的核心就是如何得到的逆过程 ，这个过程无法直接求出来，所以我们使用神经网络去拟合这一分布。我们使用一个具有参数的神经网络去计算 。假设反向的条件概率分布也是高斯分布，且高斯分布实际上只有两个参数：均值和方差，那么神经网络需要计算的实际上是

在DDPM中，方差被固定，网络只学习均值。而之后的改进模型中，方差也可由网络学习得到。

2.2.3 总结过程

总之，我们定义这么一个过程：给一张图片逐步加噪声直到变成纯粹的噪声，然后对噪声进行去噪得到真实的图片。所谓的扩散模型就是让神经网络学习这个去除噪声的方法。
所谓的加噪声，就是基于稍微干净的图片计算一个（多维）高斯分布（每个像素点都有一个高斯分布，且均值就是这个像素点的值，方差是预先定义的 ），然后从这个多维分布中抽样一个数据出来，这个数据就是加噪之后的结果。显然，如果方差非常非常小，那么每个抽样得到的像素点就和原本的像素点的值非常接近，也就是加了一个非常非常小的噪声。如果方差比较大，那么抽样结果就会和原本的结果差距较大。
去噪声也是同理，我们基于稍微噪声的图片 计算一个条件分布，我们希望从这个分布中抽样得到的是相比于 更加接近真实图片的稍微干净的图片。我们假设这样的条件分布是存在的，并且也是个高斯分布，那么我们只需要知道均值和方差就可以了。问题是这个均值和方差是无法直接计算的，所以用神经网络去学习近似这样一个高斯分布。

2.3 网络训练流程

我们最终要训练的实际上是一个噪声预测器。神经网络输出的噪声是，而真实的噪声取自于正态分布。则损失函数为：

预测网络方面，DDPM采用了 U-Net。

从而，网络的训练流程为：

我们接受一个随机的样本；

我们随机从 1 到 T 采样一个 t；

我们从高斯分布采样一些噪声并且施加在输入上；

网络从被影响过后的噪声图片学习其被施加了的噪声。

代码

3.1 Network helpers

先是一些辅助函数和类。

def exists(x):    return x is not None# 有val时返回val，val为None时返回ddef default(val, d):    if exists(val):        return val    return d() if isfunction(d) else d# 残差模块，将输入加到输出上class Residual(nn.Module):    def __init__(self, fn):        super().__init__()        self.fn = fn    def forward(self, x, *args, **kwargs):        return self.fn(x, *args, **kwargs) + x# 上采样（反卷积）def Upsample(dim):    return nn.ConvTranspose2d(dim, dim, 4, 2, 1)# 下采样def Downsample(dim):    return nn.Conv2d(dim, dim, 4, 2, 1)

3.2 Positional embeddings

类似于Transformer的positional embedding，为了让网络知道当前处理的是一系列去噪过程中的哪一个step，我们需要将步数 t 也编码并传入网络之中。DDPM采用正弦位置编码（Sinusoidal Positional Embeddings）。这一方法的输入是shape为 (batch_size, 1) 的 tensor，也就是batch中每一个sample所处的t ，并将这个tensor转换为shape为 (batch_size, dim) 的 tensor。这个tensor会被加到每一个残差模块中。

class SinusoidalPositionEmbeddings(nn.Module):    def __init__(self, dim):        super().__init__()        self.dim = dim    def forward(self, time):        device = time.device        half_dim = self.dim // 2        embeddings = math.log(10000) / (half_dim - 1)        embeddings = torch.exp(torch.arange(half_dim, device=device) * -embeddings)        embeddings = time[:, None] * embeddings[None, :]        embeddings = torch.cat((embeddings.sin(), embeddings.cos()), dim=-1)        return embeddings

3.3 ResNet/ConvNeXT block

U-Net的Block实现，可以用ResNet或ConvNeXT。

class Block(nn.Module):    def __init__(self, dim, dim_out, groups = 8):        super().__init__()        self.proj = nn.Conv2d(dim, dim_out, 3, padding = 1)        self.norm = nn.GroupNorm(groups, dim_out)        self.act = nn.SiLU()    def forward(self, x, scale_shift = None):        x = self.proj(x)        x = self.norm(x)        if exists(scale_shift):            scale, shift = scale_shift            x = x * (scale + 1) + shift        x = self.act(x)        return xclass ResnetBlock(nn.Module):    """Deep Residual Learning for Image Recognition"""        def __init__(self, dim, dim_out, *, time_emb_dim=None, groups=8):        super().__init__()        self.mlp = (            nn.Sequential(nn.SiLU(), nn.Linear(time_emb_dim, dim_out))            if exists(time_emb_dim)            else None        )        self.block1 = Block(dim, dim_out, groups=groups)        self.block2 = Block(dim_out, dim_out, groups=groups)        self.res_conv = nn.Conv2d(dim, dim_out, 1) if dim != dim_out else nn.Identity()    def forward(self, x, time_emb=None):        h = self.block1(x)        if exists(self.mlp) and exists(time_emb):            time_emb = self.mlp(time_emb)            h = rearrange(time_emb, "b c -> b c 1 1") + h        h = self.block2(h)        return h + self.res_conv(x)    class ConvNextBlock(nn.Module):    """A ConvNet for the 2020s"""    def __init__(self, dim, dim_out, *, time_emb_dim=None, mult=2, norm=True):        super().__init__()        self.mlp = (            nn.Sequential(nn.GELU(), nn.Linear(time_emb_dim, dim))            if exists(time_emb_dim)            else None        )        self.ds_conv = nn.Conv2d(dim, dim, 7, padding=3, groups=dim)        Get an email address at self.net. It's ad-free, reliable email that's based on your own name | self.net = nn.Sequential(            nn.GroupNorm(1, dim) if norm else nn.Identity(),            nn.Conv2d(dim, dim_out * mult, 3, padding=1),            nn.GELU(),            nn.GroupNorm(1, dim_out * mult),            nn.Conv2d(dim_out * mult, dim_out, 3, padding=1),        )        self.res_conv = nn.Conv2d(dim, dim_out, 1) if dim != dim_out else nn.Identity()    def forward(self, x, time_emb=None):        h = self.ds_conv(x)        if exists(self.mlp) and exists(time_emb):            condition = self.mlp(time_emb)            h = h + rearrange(condition, "b c -> b c 1 1")        h = Get an email address at self.net. It's ad-free, reliable email that's based on your own name | self.net(h)        return h + self.res_conv(x)

3.4 Attention module

包含两种attention模块，一个是常规的 multi-head self-attention，一个是 linear attention variant。

class Attention(nn.Module):    def __init__(self, dim, heads=4, dim_head=32):        super().__init__()        self.scale = dim_head**-0.5        self.heads = heads        hidden_dim = dim_head * heads        self.to_qkv = nn.Conv2d(dim, hidden_dim * 3, 1, bias=False)        self.to_out = nn.Conv2d(hidden_dim, dim, 1)    def forward(self, x):        b, c, h, w = x.shape        qkv = self.to_qkv(x).chunk(3, dim=1)        q, k, v = map(            lambda t: rearrange(t, "b (h c) x y -> b h c (x y)", h=self.heads), qkv        )        q = q * self.scale        sim = einsum("b h d i, b h d j -> b h i j", q, k)        sim = sim - sim.amax(dim=-1, keepdim=True).detach()        attn = sim.softmax(dim=-1)        out = einsum("b h i j, b h d j -> b h i d", attn, v)        out = rearrange(out, "b h (x y) d -> b (h d) x y", x=h, y=w)        return self.to_out(out)class LinearAttention(nn.Module):    def __init__(self, dim, heads=4, dim_head=32):        super().__init__()        self.scale = dim_head**-0.5        self.heads = heads        hidden_dim = dim_head * heads        self.to_qkv = nn.Conv2d(dim, hidden_dim * 3, 1, bias=False)        self.to_out = nn.Sequential(nn.Conv2d(hidden_dim, dim, 1),                                     nn.GroupNorm(1, dim))    def forward(self, x):        b, c, h, w = x.shape        qkv = self.to_qkv(x).chunk(3, dim=1)        q, k, v = map(            lambda t: rearrange(t, "b (h c) x y -> b h c (x y)", h=self.heads), qkv        )        q = q.softmax(dim=-2)        k = k.softmax(dim=-1)        q = q * self.scale        context = torch.einsum("b h d n, b h e n -> b h d e", k, v)        out = torch.einsum("b h d e, b h d n -> b h e n", context, q)        out = rearrange(out, "b h c (x y) -> b (h c) x y", h=self.heads, x=h, y=w)        return self.to_out(out)

3.5 Group normalization

DDPM的作者对U-Net的卷积/注意力层使用GN正则化。下面，我们定义了一个PreNorm类，它将被用于在注意力层之前应用groupnorm。值得注意的是，归一化在Transformer中是在注意力之前还是之后应用，目前仍存在着争议。

class PreNorm(nn.Module):    def __init__(self, dim, fn):        super().__init__()        self.fn = fn        self.norm = nn.GroupNorm(1, dim)    def forward(self, x):        x = self.norm(x)        return self.fn(x)

3.6 Conditional U-Net

现在，我们已经定义了所有的组件，接下来就是定义完整的网络了。

输入：噪声图片的batch+这些图片各自的t。

输出：预测每个图片上所添加的噪声。

Input：a batch of noisy images of shape ( batch_size, num_channels, h, w ) and a batch of steps of shape ( batch_size, 1 )
output: a tensor of shape ( batch_size, num_channels, h, w )

具体的网络结构：

首先，输入通过一个卷积层，同时计算step t 所对应的embedding

通过一系列的下采样stage，每个stage都包含：2个ResNet/ConvNeXT blocks + groupnorm + attention + residual connection + downsample operation

在网络中间，应用一个带attention的ResNet或者ConvNeXT

通过一系列的上采样stage，每个stage都包含：2个ResNet/ConvNeXT blocks + groupnorm + attention + residual connection + upsample operation

最终，通过一个ResNet/ConvNeXT blocl和一个卷积层。

class Unet(nn.Module):    def __init__(        self,        dim,        init_dim=None,        out_dim=None,        dim_mults=(1, 2, 4, 8),        channels=3,        with_time_emb=True,        resnet_block_groups=8,        use_convnext=True,        convnext_mult=2,    ):        super().__init__()        # determine dimensions        self.channels = channels        init_dim = default(init_dim, dim // 3 * 2)        self.init_conv = nn.Conv2d(channels, init_dim, 7, padding=3)        dims = [init_dim, *map(lambda m: dim * m, dim_mults)]        in_out = list(zip(dims[:-1], dims[1:]))                if use_convnext:            block_klass = partial(ConvNextBlock, mult=convnext_mult)        else:            block_klass = partial(ResnetBlock, groups=resnet_block_groups)        # time embeddings        if with_time_emb:            time_dim = dim * 4            self.time_mlp = nn.Sequential(                SinusoidalPositionEmbeddings(dim),                nn.Linear(dim, time_dim),                nn.GELU(),                nn.Linear(time_dim, time_dim),            )        else:            time_dim = None            self.time_mlp = None        # layers        self.downs = nn.ModuleList([])        self.ups = nn.ModuleList([])        num_resolutions = len(in_out)        for ind, (dim_in, dim_out) in enumerate(in_out):            is_last = ind >= (num_resolutions - 1)            self.downs.append(                nn.ModuleList(                    [                        block_klass(dim_in, dim_out, time_emb_dim=time_dim),                        block_klass(dim_out, dim_out, time_emb_dim=time_dim),                        Residual(PreNorm(dim_out, LinearAttention(dim_out))),                        Downsample(dim_out) if not is_last else nn.Identity(),                    ]                )            )        mid_dim = dims[-1]        self.mid_block1 = block_klass(mid_dim, mid_dim, time_emb_dim=time_dim)        self.mid_attn = Residual(PreNorm(mid_dim, Attention(mid_dim)))        self.mid_block2 = block_klass(mid_dim, mid_dim, time_emb_dim=time_dim)        for ind, (dim_in, dim_out) in enumerate(reversed(in_out[1:])):            is_last = ind >= (num_resolutions - 1)            self.ups.append(                nn.ModuleList(                    [                        block_klass(dim_out * 2, dim_in, time_emb_dim=time_dim),                        block_klass(dim_in, dim_in, time_emb_dim=time_dim),                        Residual(PreNorm(dim_in, LinearAttention(dim_in))),                        Upsample(dim_in) if not is_last else nn.Identity(),                    ]                )            )        out_dim = default(out_dim, channels)        self.final_conv = nn.Sequential(            block_klass(dim, dim), nn.Conv2d(dim, out_dim, 1)        )    def forward(self, x, time):        x = self.init_conv(x)        t = self.time_mlp(time) if exists(self.time_mlp) else None        h = []        # downsample        for block1, block2, attn, downsample in self.downs:            x = block1(x, t)            x = block2(x, t)            x = attn(x)            h.append(x)            x = downsample(x)        # bottleneck        x = self.mid_block1(x, t)        x = self.mid_attn(x)        x = self.mid_block2(x, t)        # upsample        for block1, block2, attn, upsample in self.ups:            x = torch.cat((x, h.pop()), dim=1)            x = block1(x, t)            x = block2(x, t)            x = attn(x)            x = upsample(x)        return self.final_conv(x)

3.7 定义前向扩散过程

DDPM中使用linear schedule定义 。后续的研究指出使用cosine schedule可能会有更好的效果。

接下来是一些简单的对于 schedule 的定义，从当中选一个使用即可。

def cosine_beta_schedule(timesteps, s=0.008):    """    cosine schedule as proposed in https://arxiv.org/abs/2102.09672    """    steps = timesteps + 1    x = torch.linspace(0, timesteps, steps)    alphas_cumprod = torch.cos(((x / timesteps) + s) / (1 + s) * torch.pi * 0.5) ** 2    alphas_cumprod = alphas_cumprod / alphas_cumprod[0]    betas = 1 - (alphas_cumprod[1:] / alphas_cumprod[:-1])    return torch.clip(betas, 0.0001, 0.9999)def linear_beta_schedule(timesteps):    beta_start = 0.0001    beta_end = 0.02    return torch.linspace(beta_start, beta_end, timesteps)def quadratic_beta_schedule(timesteps):    beta_start = 0.0001    beta_end = 0.02    return torch.linspace(beta_start**0.5, beta_end**0.5, timesteps) ** 2def sigmoid_beta_schedule(timesteps):    beta_start = 0.0001    beta_end = 0.02    betas = torch.linspace(-6, 6, timesteps)    return torch.sigmoid(betas) * (beta_end - beta_start) + beta_start

我们按照DDPM中用第二种的linear，将 T 设置为200，并将每个 t 下的各种参数提前计算好。

timesteps = 200# define beta schedulebetas = linear_beta_schedule(timesteps=timesteps)# define alphas alphas = 1. - betasalphas_cumprod = torch.cumprod(alphas, axis=0)alphas_cumprod_prev = F.pad(alphas_cumprod[:-1], (1, 0), value=1.0)sqrt_recip_alphas = torch.sqrt(1.0 / alphas)# calculations for diffusion q(x_t | x_{t-1}) and otherssqrt_alphas_cumprod = torch.sqrt(alphas_cumprod)sqrt_one_minus_alphas_cumprod = torch.sqrt(1. - alphas_cumprod)# calculations for posterior q(x_{t-1} | x_t, x_0)posterior_variance = betas * (1. - alphas_cumprod_prev) / (1. - alphas_cumprod)def extract(a, t, x_shape):    batch_size = t.shape[0]    out = a.gather(-1, t.cpu())    return out.reshape(batch_size, *((1,) * (len(x_shape) - 1))).to(t.device)

我们用一个实例来说明前向加噪过程。

from PIL import Imageimport requestsurl = 'http://images.cocodataset.org/val2017/000000039769.jpg'image = Image.open(requests.get(url, stream=True).raw)image

from torchvision.transforms import Compose, ToTensor, Lambda, ToPILImage, CenterCrop, Resizeimage_size = 128transform = Compose([    Resize(image_size),    CenterCrop(image_size),    ToTensor(), # turn into Numpy array of shape HWC, divide by 255    Lambda(lambda t: (t * 2) - 1),])x_start = transform(image).unsqueeze(0)x_start.shape  # 输出的结果是 torch.Size([1, 3, 128, 128])import numpy as npreverse_transform = Compose([     Lambda(lambda t: (t + 1) / 2),     Lambda(lambda t: t.permute(1, 2, 0)), # CHW to HWC     Lambda(lambda t: t * 255.),     Lambda(lambda t: t.numpy().astype(np.uint8)),     ToPILImage(),])

准备齐全，接下来就可以定义正向扩散过程了。

# forward diffusion (using the nice property)def q_sample(x_start, t, noise=None):    if noise is None:        noise = torch.randn_like(x_start)    sqrt_alphas_cumprod_t = extract(sqrt_alphas_cumprod, t, x_start.shape)    sqrt_one_minus_alphas_cumprod_t = extract(        sqrt_one_minus_alphas_cumprod, t, x_start.shape    )    return sqrt_alphas_cumprod_t * x_start + sqrt_one_minus_alphas_cumprod_t * noisedef get_noisy_image(x_start, t):  # add noise  x_noisy = q_sample(x_start, t=t)  # turn back into PIL image  noisy_image = reverse_transform(x_noisy.squeeze())  return noisy_image

可视化一下多个不同t的生成结果。

import matplotlib.pyplot as plt# use seed for reproducabilitytorch.manual_seed(0)# source: https://pytorch.org/vision/stable/auto_examples/plot_transforms.html#sphx-glr-auto-examples-plot-transforms-pydef plot(imgs, with_orig=False, row_title=None, **imshow_kwargs):    if not isinstance(imgs[0], list):        # Make a 2d grid even if there's just 1 row        imgs = [imgs]    num_rows = len(imgs)    num_cols = len(imgs[0]) + with_orig    fig, axs = plt.subplots(figsize=(200,200), nrows=num_rows, ncols=num_cols, squeeze=False)    for row_idx, row in enumerate(imgs):        row = [image] + row if with_orig else row        for col_idx, img in enumerate(row):            ax = axs[row_idx, col_idx]            ax.imshow(np.asarray(img), **imshow_kwargs)            ax.set(xticklabels=[], yticklabels=[], xticks=[], yticks=[])    if with_orig:        axs[0, 0].set(title='Original image')        axs[0, 0].title.set_size(8)    if row_title is not None:        for row_idx in range(num_rows):            axs[row_idx, 0].set(ylabel=row_title[row_idx])    plt.tight_layout()plot([get_noisy_image(x_start, torch.tensor([t])) for t in [0, 50, 100, 150, 199]])

3.8 定义损失函数

def p_losses(denoise_model, x_start, t, noise=None, loss_type="l1"):    # 先采样噪声    if noise is None:        noise = torch.randn_like(x_start)        # 用采样得到的噪声去加噪图片    x_noisy = q_sample(x_start=x_start, t=t, noise=noise)    predicted_noise = denoise_model(x_noisy, t)        # 根据加噪了的图片去预测采样的噪声    if loss_type == 'l1':        loss = F.l1_loss(noise, predicted_noise)    elif loss_type == 'l2':        loss = F.mse_loss(noise, predicted_noise)    elif loss_type == "huber":        loss = F.smooth_l1_loss(noise, predicted_noise)    else:        raise NotImplementedError()    return loss

3.9 定义数据集 PyTorch Dataset 和 DataLoader

我们使用mnist数据集构造了一个 DataLoader，每个batch由128张 normalize 过的 image 组成。

from datasets import load_dataset# load dataset from the hubdataset = load_dataset("fashion_mnist")image_size = 28channels = 1batch_size = 128from torchvision import transformsfrom torch.utils.data import DataLoadertransform = Compose([            transforms.RandomHorizontalFlip(),            transforms.ToTensor(),            transforms.Lambda(lambda t: (t * 2) - 1)])def transforms(examples):   examples["pixel_values"] = [transform(image.convert("L")) for image in examples["image"]]   del examples["image"]   return examplestransformed_dataset = dataset.with_transform(transforms).remove_columns("label")dataloader = DataLoader(transformed_dataset["train"], batch_size=batch_size, shuffle=True)batch = next(iter(dataloader))print(batch.keys())    # dict_keys(['pixel_values'])

3.10 采样

采样过程发生在反向去噪时。对于一张纯噪声，扩散模型一步步地去除噪声最终得到真实图片，采样事实上就是定义的去除噪声这一行为。 观察采样算法中第四行， t−1 步的图片是由 t 步的图片减去一个噪声得到的，只不过这个噪声是由网络拟合出来，并且 rescale 过的而已。 这里要注意第四行式子的最后一项，采样时每一步也都会加上一个从正态分布采样的纯噪声。理想情况下，最终我们会得到一张看起来像是从真实数据分布中采样得到的图片。

@torch.no_grad()def p_sample(model, x, t, t_index):    betas_t = extract(betas, t, x.shape)    sqrt_one_minus_alphas_cumprod_t = extract(        sqrt_one_minus_alphas_cumprod, t, x.shape    )    sqrt_recip_alphas_t = extract(sqrt_recip_alphas, t, x.shape)        # Equation 11 in the paper    # Use our model (noise predictor) to predict the mean    model_mean = sqrt_recip_alphas_t * (        x - betas_t * model(x, t) / sqrt_one_minus_alphas_cumprod_t    )    if t_index == 0:        return model_mean    else:        posterior_variance_t = extract(posterior_variance, t, x.shape)        noise = torch.randn_like(x)        # Algorithm 2 line 4:        return model_mean + torch.sqrt(posterior_variance_t) * noise # Algorithm 2 (including returning all images)@torch.no_grad()def p_sample_loop(model, shape):    device = next(model.parameters()).device    b = shape[0]    # start from pure noise (for each example in the batch)    img = torch.randn(shape, device=device)    imgs = []    for i in tqdm(reversed(range(0, timesteps)), desc='sampling loop time step', total=timesteps):        img = p_sample(model, img, torch.full((b,), i, device=device, dtype=torch.long), i)        imgs.append(img.cpu().numpy())    return imgs@torch.no_grad()def sample(model, image_size, batch_size=16, channels=3):    return p_sample_loop(model, shape=(batch_size, channels, image_size, image_size))

3.11 训练

先定义一些辅助生成图片的函数。

from pathlib import Pathdef num_to_groups(num, divisor):    groups = num // divisor    remainder = num % divisor    arr = [divisor] * groups    if remainder > 0:        arr.append(remainder)    return arrresults_folder = Path("./results")results_folder.mkdir(exist_ok = True)save_and_sample_every = 1000

接下来实例化模型。

from torch.optim import Adamdevice = "cuda" if torch.cuda.is_available() else "cpu"model = Unet(    dim=image_size,    channels=channels,    dim_mults=(1, 2, 4,))model.to(device)optimizer = Adam(model.parameters(), lr=1e-3)

开始训练！

from torchvision.utils import save_imageepochs = 6for epoch in range(epochs):    for step, batch in enumerate(dataloader):      optimizer.zero_grad()      batch_size = batch["pixel_values"].shape[0]      batch = batch["pixel_values"].to(device)      # Algorithm 1 line 3: sample t uniformally for every example in the batch      t = torch.randint(0, timesteps, (batch_size,), device=device).long()      loss = p_losses(model, batch, t, loss_type="huber")      if step % 100 == 0:        print("Loss:", loss.item())      loss.backward()      optimizer.step()      # save generated images      if step != 0 and step % save_and_sample_every == 0:        milestone = step // save_and_sample_every        batches = num_to_groups(4, batch_size)        all_images_list = list(map(lambda n: sample(model, batch_size=n, channels=channels), batches))        all_images = torch.cat(all_images_list, dim=0)        all_images = (all_images + 1) * 0.5        save_image(all_images, str(results_folder / f'sample-{milestone}.png'), nrow = 6)

Inference：

# sample 64 imagessamples = sample(model, image_size=image_size, batch_size=64, channels=channels)# show a random onerandom_index = 5plt.imshow(samples[-1][random_index].reshape(image_size, image_size, channels), cmap="gray")

import matplotlib.animation as animationrandom_index = 53fig = plt.figure()ims = []for i in range(timesteps):    im = plt.imshow(samples[i][random_index].reshape(image_size, image_size, channels), cmap="gray", animated=True)    ims.append([im])animate = animation.ArtistAnimation(fig, ims, interval=50, blit=True, repeat_delay=1000)animate.save('diffusion.gif')plt.show()

4. 参考文献

原理+代码：Diffusion Model 直观理解

The Annotated Diffusion Model

【diffusion】扩散模型详解！理论＋代码