什么是扩散模型

Diffusion model-based inverse design for thermal transparency

本篇论文是关于使用扩散模型对热透明度的逆向设计内容的解读。

首先什么是扩散模型：
扩散模型的核心思想是将数据从一个已知分布逐渐转换到一个简单的目标分布（通常是高斯分布），然后再逆转这个过程，从简单的分布中生成数据。这个过程包括两个主要阶段：正向过程（前向扩散）和反向过程（逆向扩散）。
扩散模型受到非平衡热力学的启发。 它们定义了一个马尔可夫扩散步骤链，将随机噪音缓慢添加到数据中，然后学习逆向扩散过程，从噪音中构建所需的数据样本。
通俗来讲就是，我们给定一个样本x0，并且由x0与t个高斯白噪声的线性组合最后形成了xt(也就是一张什么都没有的噪声图）。这就是前向扩散过程。

用公式来表示则是以下：由图片最后的公式可以看出来，在前向加噪的过程中我们可以实现一步加噪成功。最后用q(xt|x0)来表示x0到xt的正态分布。


图片上随着线性组合的次数增加，值越来越小，是说明图片本身的内容占比在减小。
接下来加上扩散模型的逆向过程。

我们可以通过已知的xt和预测的高斯噪声 ϵ 来推导出 x(t−1)。在逆向过程中通常需要迭代进行，每一步去噪都依赖于前一步的结果，并且通过神经网络来预测当前步骤的噪声 。
那么如何实现预测去噪的高斯白噪声 ϵ的值呢。这时便引入q(xt-1|xt，x0)。
（具体的计算过程可以自己去算一遍：https://lilianweng.github.io/posts/2021-07-11-diffusion-models/#:~:text=Diffusion%20models%20are%20inspired%20by,data%20samples%20from%20the%20noise）
最终的计算公式为上图。
预测的思路是，通过大概的预测x0，得出要去的噪声，然后通过大量的预测，一点一点的还原出想要的样本图，并且使用迭代预测的方法还可以不断的进行修正。
接下来是我找到的训练示例（项目来源于：https://github.com/TeaPearce/Conditional_Diffusion_MNIST）

''' 
This script does conditional image generation on MNIST, using a diffusion model

This code is modified from,
https://github.com/cloneofsimo/minDiffusion

Diffusion model is based on DDPM,
https://arxiv.org/abs/2006.11239

The conditioning idea is taken from 'Classifier-Free Diffusion Guidance',
https://arxiv.org/abs/2207.12598

This technique also features in ImageGen 'Photorealistic Text-to-Image Diffusion Modelswith Deep Language Understanding',
https://arxiv.org/abs/2205.11487

'''

from typing import Dict, Tuple
from tqdm import tqdm
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.data import DataLoader
from torchvision import models, transforms
from torchvision.datasets import MNIST
from torchvision.utils import save_image, make_grid
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation, PillowWriter
import numpy as np

class ResidualConvBlock(nn.Module):
    def __init__(
        self, in_channels: int, out_channels: int, is_res: bool = False
    ) -> None:
        super().__init__()
        '''
        standard ResNet style convolutional block
        '''
        self.same_channels = in_channels==out_channels
        self.is_res = is_res
        self.conv1 = nn.Sequential(
            nn.Conv2d(in_channels, out_channels, 3, 1, 1),
            nn.BatchNorm2d(out_channels),
            nn.GELU(),
        )
        self.conv2 = nn.Sequential(
            nn.Conv2d(out_channels, out_channels, 3, 1, 1),
            nn.BatchNorm2d(out_channels),
            nn.GELU(),
        )

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        if self.is_res:
            x1 = self.conv1(x)
            x2 = self.conv2(x1)
            # this adds on correct residual in case channels have increased
            if self.same_channels:
                out = x + x2
            else:
                out = x1 + x2 
            return out / 1.414
        else:
            x1 = self.conv1(x)
            x2 = self.conv2(x1)
            return x2


class UnetDown(nn.Module):
    def __init__(self, in_channels, out_channels):
        super(UnetDown, self).__init__()
        '''
        process and downscale the image feature maps
        '''
        layers = [ResidualConvBlock(in_channels, out_channels), nn.MaxPool2d(2)]
        self.model = nn.Sequential(*layers)

    def forward(self, x):
        return self.model(x)


class UnetUp(nn.Module):
    def __init__(self, in_channels, out_channels):
        super(UnetUp, self).__init__()
        '''
        process and upscale the image feature maps
        '''
        layers = [
            nn.ConvTranspose2d(in_channels, out_channels, 2, 2),
            ResidualConvBlock(out_channels, out_channels),
            ResidualConvBlock(out_channels, out_channels),
        ]
        self.model = nn.Sequential(*layers)

    def forward(self, x, skip):
        x = torch.cat((x, skip), 1)
        x = self.model(x)
        return x


class EmbedFC(nn.Module):
    def __init__(self, input_dim, emb_dim):
        super(EmbedFC, self).__init__()
        '''
        generic one layer FC NN for embedding things  
        '''
        self.input_dim = input_dim
        layers = [
            nn.Linear(input_dim, emb_dim),
            nn.GELU(),
            nn.Linear(emb_dim, emb_dim),
        ]
        self.model = nn.Sequential(*layers)

    def forward(self, x):
        x = x.view(-1, self.input_dim)
        return self.model(x)


class ContextUnet(nn.Module):
    def __init__(self, in_channels, n_feat = 256, n_classes=10):
        super(ContextUnet, self).__init__()

        self.in_channels = in_channels
        self.n_feat = n_feat
        self.n_classes = n_classes

        self.init_conv = ResidualConvBlock(in_channels, n_feat, is_res=True)

        self.down1 = UnetDown(n_feat, n_feat)
        self.down2 = UnetDown(n_feat, 2 * n_feat)

        self.to_vec = nn.Sequential(nn.AvgPool2d(7), nn.GELU())

        self.timeembed1 = EmbedFC(1, 2*n_feat)
        self.timeembed2 = EmbedFC(1, 1*n_feat)
        self.contextembed1 = EmbedFC(n_classes, 2*n_feat)
        self.contextembed2 = EmbedFC(n_classes, 1*n_feat)

        self.up0 = nn.Sequential(
            # nn.ConvTranspose2d(6 * n_feat, 2 * n_feat, 7, 7), # when concat temb and cemb end up w 6*n_feat
            nn.ConvTranspose2d(2 * n_feat, 2 * n_feat, 7, 7), # otherwise just have 2*n_feat
            nn.GroupNorm(8, 2 * n_feat),
            nn.ReLU(),
        )

        self.up1 = UnetUp(4 * n_feat, n_feat)
        self.up2 = UnetUp(2 * n_feat, n_feat)
        self.out = nn.Sequential(
            nn.Conv2d(2 * n_feat, n_feat, 3, 1, 1),
            nn.GroupNorm(8, n_feat),
            nn.ReLU(),
            nn.Conv2d(n_feat, self.in_channels, 3, 1, 1),
        )

    def forward(self, x, c, t, context_mask):
        # x is (noisy) image, c is context label, t is timestep, 
        # context_mask says which samples to block the context on

        x = self.init_conv(x)
        down1 = self.down1(x)
        down2 = self.down2(down1)
        hiddenvec = self.to_vec(down2)

        # convert context to one hot embedding
        c = nn.functional.one_hot(c, num_classes=self.n_classes).type(torch.float)
        
        # mask out context if context_mask == 1
        context_mask = context_mask[:, None]
        context_mask = context_mask.repeat(1,self.n_classes)
        context_mask = (-1*(1-context_mask)) # need to flip 0 <-> 1
        c = c * context_mask
        
        # embed context, time step
        cemb1 = self.contextembed1(c).view(-1, self.n_feat * 2, 1, 1)
        temb1 = self.timeembed1(t).view(-1, self.n_feat * 2, 1, 1)
        cemb2 = self.contextembed2(c).view(-1, self.n_feat, 1, 1)
        temb2 = self.timeembed2(t).view(-1, self.n_feat, 1, 1)

        # could concatenate the context embedding here instead of adaGN
        # hiddenvec = torch.cat((hiddenvec, temb1, cemb1), 1)

        up1 = self.up0(hiddenvec)
        # up2 = self.up1(up1, down2) # if want to avoid add and multiply embeddings
        up2 = self.up1(cemb1*up1+ temb1, down2)  # add and multiply embeddings
        up3 = self.up2(cemb2*up2+ temb2, down1)
        out = self.out(torch.cat((up3, x), 1))
        return out


def ddpm_schedules(beta1, beta2, T):
    """
    Returns pre-computed schedules for DDPM sampling, training process.
    """
    assert beta1 < beta2 < 1.0, "beta1 and beta2 must be in (0, 1)"

    beta_t = (beta2 - beta1) * torch.arange(0, T + 1, dtype=torch.float32) / T + beta1
    sqrt_beta_t = torch.sqrt(beta_t)
    alpha_t = 1 - beta_t
    log_alpha_t = torch.log(alpha_t)
    alphabar_t = torch.cumsum(log_alpha_t, dim=0).exp()

    sqrtab = torch.sqrt(alphabar_t)
    oneover_sqrta = 1 / torch.sqrt(alpha_t)

    sqrtmab = torch.sqrt(1 - alphabar_t)
    mab_over_sqrtmab_inv = (1 - alpha_t) / sqrtmab

    return {
        "alpha_t": alpha_t,  # \alpha_t
        "oneover_sqrta": oneover_sqrta,  # 1/\sqrt{\alpha_t}
        "sqrt_beta_t": sqrt_beta_t,  # \sqrt{\beta_t}
        "alphabar_t": alphabar_t,  # \bar{\alpha_t}
        "sqrtab": sqrtab,  # \sqrt{\bar{\alpha_t}}
        "sqrtmab": sqrtmab,  # \sqrt{1-\bar{\alpha_t}}
        "mab_over_sqrtmab": mab_over_sqrtmab_inv,  # (1-\alpha_t)/\sqrt{1-\bar{\alpha_t}}
    }


class DDPM(nn.Module):
    def __init__(self, nn_model, betas, n_T, device, drop_prob=0.1):
        super(DDPM, self).__init__()
        self.nn_model = nn_model.to(device)

        # register_buffer allows accessing dictionary produced by ddpm_schedules
        # e.g. can access self.sqrtab later
        for k, v in ddpm_schedules(betas[0], betas[1], n_T).items():
            self.register_buffer(k, v)

        self.n_T = n_T
        self.device = device
        self.drop_prob = drop_prob
        self.loss_mse = nn.MSELoss()

    def forward(self, x, c):
        """
        this method is used in training, so samples t and noise randomly
        """

        _ts = torch.randint(1, self.n_T+1, (x.shape[0],)).to(self.device)  # t ~ Uniform(0, n_T)
        noise = torch.randn_like(x)  # eps ~ N(0, 1)

        x_t = (
            self.sqrtab[_ts, None, None, None] * x
            + self.sqrtmab[_ts, None, None, None] * noise
        )  # This is the x_t, which is sqrt(alphabar) x_0 + sqrt(1-alphabar) * eps
        # We should predict the "error term" from this x_t. Loss is what we return.

        # dropout context with some probability
        context_mask = torch.bernoulli(torch.zeros_like(c)+self.drop_prob).to(self.device)
        
        # return MSE between added noise, and our predicted noise
        return self.loss_mse(noise, self.nn_model(x_t, c, _ts / self.n_T, context_mask))

    def sample(self, n_sample, size, device, guide_w = 0.0):
        # we follow the guidance sampling scheme described in 'Classifier-Free Diffusion Guidance'
        # to make the fwd passes efficient, we concat two versions of the dataset,
        # one with context_mask=0 and the other context_mask=1
        # we then mix the outputs with the guidance scale, w
        # where w>0 means more guidance

        x_i = torch.randn(n_sample, *size).to(device)  # x_T ~ N(0, 1), sample initial noise
        c_i = torch.arange(0,10).to(device) # context for us just cycles throught the mnist labels
        c_i = c_i.repeat(int(n_sample/c_i.shape[0]))

        # don't drop context at test time
        context_mask = torch.zeros_like(c_i).to(device)

        # double the batch
        c_i = c_i.repeat(2)
        context_mask = context_mask.repeat(2)
        context_mask[n_sample:] = 1. # makes second half of batch context free

        x_i_store = [] # keep track of generated steps in case want to plot something 
        print()
        for i in range(self.n_T, 0, -1):
            print(f'sampling timestep {i}',end='\r')
            t_is = torch.tensor([i / self.n_T]).to(device)
            t_is = t_is.repeat(n_sample,1,1,1)

            # double batch
            x_i = x_i.repeat(2,1,1,1)
            t_is = t_is.repeat(2,1,1,1)

            z = torch.randn(n_sample, *size).to(device) if i > 1 else 0

            # split predictions and compute weighting
            eps = self.nn_model(x_i, c_i, t_is, context_mask)
            eps1 = eps[:n_sample]
            eps2 = eps[n_sample:]
            eps = (1+guide_w)*eps1 - guide_w*eps2
            x_i = x_i[:n_sample]
            x_i = (
                self.oneover_sqrta[i] * (x_i - eps * self.mab_over_sqrtmab[i])
                + self.sqrt_beta_t[i] * z
            )
            if i%20==0 or i==self.n_T or i<8:
                x_i_store.append(x_i.detach().cpu().numpy())
        
        x_i_store = np.array(x_i_store)
        return x_i, x_i_store


def train_mnist():

    # hardcoding these here
    n_epoch = 20
    batch_size = 256
    n_T = 400 # 500
    device = "cuda:0"
    n_classes = 10
    n_feat = 128 # 128 ok, 256 better (but slower)
    lrate = 1e-4
    save_model = False
    save_dir = 'D:/Huicheng/Desktop/神经网络设计结构色/Conditional_Diffusion_MNIST-main/data/diffusion_outputs10/'
    ws_test = [0.0, 0.5, 2.0] # strength of generative guidance

    ddpm = DDPM(nn_model=ContextUnet(in_channels=1, n_feat=n_feat, n_classes=n_classes), betas=(1e-4, 0.02), n_T=n_T, device=device, drop_prob=0.1)
    ddpm.to(device)

    # optionally load a model
    # ddpm.load_state_dict(torch.load("./data/diffusion_outputs/ddpm_unet01_mnist_9.pth"))

    tf = transforms.Compose([transforms.ToTensor()]) # mnist is already normalised 0 to 1

    dataset = MNIST("./data", train=True, download=True, transform=tf)
    dataloader = DataLoader(dataset, batch_size=batch_size, shuffle=True, num_workers=5)
    optim = torch.optim.Adam(ddpm.parameters(), lr=lrate)

    for ep in range(n_epoch):
        print(f'epoch {ep}')
        ddpm.train()

        # linear lrate decay
        optim.param_groups[0]['lr'] = lrate*(1-ep/n_epoch)

        pbar = tqdm(dataloader)
        loss_ema = None
        for x, c in pbar:
            optim.zero_grad()
            x = x.to(device)
            c = c.to(device)
            loss = ddpm(x, c)
            loss.backward()
            if loss_ema is None:
                loss_ema = loss.item()
            else:
                loss_ema = 0.95 * loss_ema + 0.05 * loss.item()
            pbar.set_description(f"loss: {loss_ema:.4f}")
            optim.step()
        
        # for eval, save an image of currently generated samples (top rows)
        # followed by real images (bottom rows)
        ddpm.eval()
        with torch.no_grad():
            n_sample = 4*n_classes
            for w_i, w in enumerate(ws_test):
                x_gen, x_gen_store = ddpm.sample(n_sample, (1, 28, 28), device, guide_w=w)

                # append some real images at bottom, order by class also
                x_real = torch.Tensor(x_gen.shape).to(device)
                for k in range(n_classes):
                    for j in range(int(n_sample/n_classes)):
                        try: 
                            idx = torch.squeeze((c == k).nonzero())[j]
                        except:
                            idx = 0
                        x_real[k+(j*n_classes)] = x[idx]

                x_all = torch.cat([x_gen, x_real])
                grid = make_grid(x_all*-1 + 1, nrow=10)
                save_image(grid, save_dir + f"image_ep{ep}_w{w}.png")
                print('saved image at ' + save_dir + f"image_ep{ep}_w{w }.png")

                if ep%5==0 or ep == int(n_epoch-1):
                    # create gif of images evolving over time, based on x_gen_store
                    fig, axs = plt.subplots(nrows=int(n_sample/n_classes), ncols=n_classes,sharex=True,sharey=True,figsize=(8,3))
                    def animate_diff(i, x_gen_store):
                        print(f'gif animating frame {i} of {x_gen_store.shape[0]}', end='\r')
                        plots = []
                        for row in range(int(n_sample/n_classes)):
                            for col in range(n_classes):
                                axs[row, col].clear()
                                axs[row, col].set_xticks([])
                                axs[row, col].set_yticks([])
                                # plots.append(axs[row, col].imshow(x_gen_store[i,(row*n_classes)+col,0],cmap='gray'))
                                plots.append(axs[row, col].imshow(-x_gen_store[i,(row*n_classes)+col,0],cmap='gray',vmin=(-x_gen_store[i]).min(), vmax=(-x_gen_store[i]).max()))
                        return plots
                    ani = FuncAnimation(fig, animate_diff, fargs=[x_gen_store],  interval=200, blit=False, repeat=True, frames=x_gen_store.shape[0])    
                    ani.save(save_dir + f"gif_ep{ep}_w{w}.gif", dpi=100, writer=PillowWriter(fps=5))
                    print('saved image at ' + save_dir + f"gif_ep{ep}_w{w}.gif")
        # optionally save model
        if save_model and ep == int(n_epoch-1):
            torch.save(ddpm.state_dict(), save_dir + f"model_{ep}.pth")
            print('saved model at ' + save_dir + f"model_{ep}.pth")

if __name__ == "__main__":
    train_mnist()

在进过20轮的周期训练生成的模型样本动图：

现在通过一个图片的样本可以通过扩散模型的逆过程将一个高斯白噪声图还原成与原样本相似的图。那么是否可以通过扩散模型进行纳米粒子的逆向设计，通过大量的纳米结构数据集来训练一个模型，再使用模型来训练扩散模型的反向设计呢？
在扩散模型反向设计热透明度中有表示：
真正具有挑战性的部分是设计神经网络的体系结构，以预测噪声€θ(xt,t)。图像和视频生成扩散模型通常从工业标准2D和3D U-Net体系结构开始。受U-Net架构的启发，通过使用一系列线性层和ReLU激活（10-20-ReLU-20-ReLU-6-ReLU）来创建六个维度的瓶颈。通过这种方法就能很好的将设计参数转化为六纬设计参数，并且保证信息损失最少。
在训练噪声预测神经网络€θ(xt,t）时的一个重要任务是评估训练过程何时收敛。在传统的神经网络训练任务中，如图像分类，损失值是训练过程收敛性的良好指标。但是，对于扩散模型的训练，损失不再是收敛的好指标。在损失迅速下降的最初几个时期之后，无论轮数如何训练，损失的价值都不会显示出明显的下降趋势。实际上，噪声预测神经网络E6(x:,t)的性能在这段时间里仍在提高。因此，我们必须找到另一种方法来量化噪声预测的神经网络€θ(x,t)的性能，作为训练的轮数的函数，以评估训练过程是否已经收敛。
我们的数值实验表明，在测试集上评估的推理误差可以很好地替代损失作为指标。更具体地说，对于每训练200个epoch，我们保存一个预测神经网络的噪声快照。然后，我们使用对测试集中随机选择的100个实例（测试集总共有5000个实例），以简化的响应向量作为条件进行推理过程。对于选择的100个实例中的每个实例，推理过程为执行了1000次，并且1000个推断值中的每个值都是平方误差的总和（称为推断误差）。然后，我们通过计算每个实例的1000个推理误差的平均值来与相应的设计参数进行对比。现在，对于预测神经网络 E8(xt,t)的噪声快照，我们有100个平均值，并且计算100个平均值的均值和方差，并将其绘制为训练轮数的函数。如图所示，推断误差的均值和方差在8000个epochs后停止下降，这表明噪声预测神经网络ø(xt,t)的训练过程已经收敛。
总结来说就是我们要先制作一个纳米结构数据集，分别为训练集和测试集，训练集用来训练模型，测试集则用于对参数的调整。进过上述的训练与测试得出预测神经网络模型。
接下来的学习目标：首先学习扩散模型的代码编写，如何将公式转化为代码，其次就是学习FDTD，理解纳米结构，最终将纳米结构的参数与代码结合训练出模型，然后实现逆向设计。