A. Encoding Network of PFSPNet The encoding network is divided into three parts. In the part I, RNN is adopted to model the processing time pij of job i on all machines, which can be converted into a fixed dimensional vector pi. In the part II, the number of machines m is integrated into the vector pi through the fully connected layer, and the fixed dimensional vector p˜i is output. In the part III, p˜i is fed into the convolution layer to improve the expression ability of the network, and the final output η p= [ η p1, η p2,..., η pn] is obtained. Fig. 2 illustrates the encoding network. In the part I, the modelling process for pij is described as follows, where WB, hij , h0 are k-dimensional vectors, h0, U, W, b and WB are the network parameters, and f() is the mapping from RNN input to hidden layer output. The main steps of the part I are shown as follows. Step 1: Input pij to the embedding layer and then obtain the output yij = WB pij ; Step 2: Input yi1 and h0 to the RNN and then obtain the hidden layer output hi1 = f(yi1,h0; U,W, b). Let p1 = h1m ; Step 3: Input yij and hi,j−1, j = 2, 3 ··· , m into RNN in turn, and then obtain the hidden layer output hij = f(yij ,hi,j−1; U,W, b), j = 2, 3 ··· , m. Let pi = him . In the part II, the number of machines m and the vector pi are integrated by the fully connected layer. The details are described as follows. WB and h˜i are d-dimensional vectors, WB W and ˜b are network parameters, and g() denotes the mapping from the input to the output of full connection layer. Step 1: Input the number of machines m to the embedding layer, and the output m = WB m is obtained。Step 2: Input m and pi to the fully connected layer and then obtain the output hi = g([m, pi];W, b); Step 3: Let pi = Relu(hi). In the part III, pi, i = 1, 2,...,n are input into onedimensional convolution layer. The final output vector η pi, i = 1, 2, ··· , n are obtained after the output of convolutional layer goes through the Relu layer.首先逐行仔细的分析此过程，其次怎么使用pytorch用EncoderNetwork类完全实现这个过程的所有功能和步骤

首先，我们需要定义一个EncoderNetwork类，它应该包含三个部分：RNN模型、全连接层和卷积层。以下是一个简单的实现：

import torch
import torch.nn as nn

class EncoderNetwork(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super(EncoderNetwork, self).__init__()

        # RNN Model
        self.rnn = nn.RNN(input_size, hidden_size, batch_first=True)

        # Fully Connected Layer
        self.fc = nn.Linear(hidden_size + 1, output_size)

        # Convolutional Layer
        self.conv1d = nn.Conv1d(output_size, output_size, kernel_size=1)

        # ReLU activation function
        self.relu = nn.ReLU()

    def forward(self, x, m):
        # Part I: RNN Model
        y = x.view(x.size(0), -1)
        h0 = torch.zeros(1, x.size(0), self.rnn.hidden_size).to(x.device)
        out, _ = self.rnn(y, h0)
        p = out[:, -1, :]

        # Part II: Fully Connected Layer
        m = m.view(-1, 1)
        h = self.fc(torch.cat((p, m), dim=1))
        p_ = self.relu(h)

        # Part III: Convolutional Layer
        p_ = p_.permute(0, 2, 1)
        p_ = self.conv1d(p_)
        p_ = self.relu(p_)
        p_ = p_.permute(0, 2, 1)

        return p_

在上面的代码中，我们定义了一个EncoderNetwork类，它有三个部分：RNN模型、全连接层和卷积层。在forward函数中，我们首先输入pij的张量x和机器数量m，然后按照Part I、Part II和Part III的顺序进行处理。最后，我们返回最终的输出张量。

在这个类的构造函数中，我们定义了RNN模型、全连接层、卷积层和ReLU激活函数。在forward函数中，我们首先将输入x转换为一个2D张量，并将h0初始化为全零张量。然后我们使用RNN模型处理x，得到输出out。在这个过程中，我们只需要使用out的最后一个时间步，即out[:,-1,:]，作为RNN模型的输出p。接下来，我们将机器数量m与p连接起来，然后将它们输入到全连接层中。最后，我们将全连接层的输出张量输入到卷积层中，并经过ReLU激活函数处理。

在使用这个类时，我们需要传入三个参数：输入大小input_size、隐藏层大小hidden_size和输出大小output_size。然后，我们可以使用encoder = EncoderNetwork(input_size, hidden_size, output_size)来创建一个EncoderNetwork对象。最后，我们可以使用encoder.forward(x, m)来计算x和m的输出张量。