## 3 Methodology
### Architecture of the Conv1D Network  
Figures 3 and 4 provide illustrations of the network architecture and the operators used in each hidden layer. The model mainly consists of five groups of conv-blocks and the fully connected part. A conv-block can be subdivided into a convolutional layer and a linear layer. When the training data are input into the convolutional layer, the convolution operation [35] is conducted. The convolution kernel scans through the input data and derives the eigenvalues, which can be expressed as:  
\[a^{l}(l_{out},c_{out})=w^{l}(c_{out},k)*a^{l-1}(l_{in},c_{in})+b^{l} \tag{4}\]  
Figure 4: Schematic diagram of operators in (**a**) convolution layer, (**b**) max pooling layer, and (**c**) linear layer. Figure 3: The detailed structure of the proposed Conv1D network. Figure 2: The general workflow of the dynamic training method. The * mark stands for changes in the datasets. where \(w^{l}\) is the weight, \(k\) is the kernel size, and \(c_{out}\) is the number of output channels. \(a^{l-1}\) is the input, \(l_{in}\) is the length of the input, \(c_{in}\) is the input channels, and \(b\) is the bias. The length of the output, \(l_{out}\), is computed from \(l_{in}\) and \(k\) using Equation (5). \[l_{out}=\frac{l_{in}+2p-k}{s}+1 \tag{5}\]  
where \(p\) is the number of zero-padding and s is the size of stride. In our model, \(p=2\) and s = 1. The convolutional layer is followed by a pooling layer, which amplifies the data features following the logic of max-pooling [36]. Unlike the convolutional layer, there are no weights and biases in the pooling layer, only the input data are scanned, and the maximum value of each kernel is selected. After the convolution operator, the ReLU nonlinear operator is performed. The nonlinear operation is calculated as:  
\[a^{l}=\max\left(0,a^{l-1}\right) \tag{6}\]  
The inclusion of nonlinear computation divides the convolutional layer and the linear layer, avoiding the phenomenon of zero grad transmission during the training stage. At the linear layer, we borrowed the idea of the fully connected neural network to build the structure. The transmission of each value is calculated using the weights and biases, then passed to the next layer after the nonlinear operator, which is calculated in Equation (7). \[a^{l}_{i}=\sigma\left(w^{l}_{i}a^{l-1}+b^{l}\right) \tag{7}\]  
where \(a^{l}_{i}\) is the ith element of the output, \(w^{l}\) is the weight of the current layer, and \(\sigma(. )\) is the nonlinear operator. The role of the fully connected layer is to fine-tune the output values from conv-blocks. Up to this point, a complete forward propagation is computed and prevents the output values from being negative, where the temperature distribution will not be lower than the ambient temperature. The fully connected layers of the model consist of two linear layers. The nonlinear activation of the last layer is a sigmoid function (Equation (8)). \[a^{l}=\frac{1}{1+\exp\left(-a^{l-1}\right)} \tag{8}\]  
One complete propagation of the model includes forward propagation and backpropagation. In our model, inputting the data of length 51 will obtain the predicted temperature field data \(\hat{T}\) of size 51 \(\times\) 51 after implementing one forward propagation. The detailed size of each layer's output is shown in Figure 3. Then, the model will calculate the loss function and implement back-propagation. The propagation gradient according to the learning rate will be calculated to update the parameters of the hidden layer in the stage; this process is also called regression or gradient descent. The mean squared error (MSE) function is chosen as the loss function for the regression stage. Equation (9) defines the MSE loss function. \[MSE=\frac{1}{N}~{}\sum_{1}^{N}\left[\hat{T}(x,y)-T(x,y)\right]^{2} \tag{9}\]  
where \(T\) is the original temperature data in the training set and \(\hat{T}\) is the output matrix of the model, which represents the predicted temperature. \(N\) is the total number of training samples.