# Prediction of Corroded Pipeline Failure Pressure Based on Empirical Knowledge and Machine Learning
## 3 Training Details  
### Dataset  
The dataset employed in this study is derived from the comprehensive compilation presented in Reference [19]. This dataset is notable for its diversity, as it integrates 60 burst test results from various literature sources, corresponding to distinct defect conditions in pipelines. These conditions include different corrosion depths, lengths, and widths, reflecting a wide range of realistic scenarios encountered in the assessment of corroded pipelines. The heterogeneity of the dataset, with its varied defect geometries and material properties, poses significant challenges for traditional assessment methods. However, the method proposed in this research demonstrates robust tolerance to these diverse defect conditions. This capability is crucial for accurately predicting the failure pressure of pipelines under complex corrosion scenarios, thereby enhancing the reliability and safety of pipeline integrity assessments. The specifications of 60 samples extracted from the relevant literature are presented in Table 1[20, 21, 22, 23, 24, 25]. The dataset was divided into training and validation sets at a ratio of 8:2. To balance the data distribution between the training and validation sets, the data samples among validation set were evenly selected from various sources. Figure 1: Empirical formula-constrained optimization for predicting failure pressure in corroded pipelines. to be 32. Each layer of the network consists of three components: a linear module, a group normalization module [26], and a ReLU activation function. The linear module is responsible for the initial transformation of the input data, while the group normalization module helps to stabilize the training process by normalizing the data across groups of channels. Finally, the ReLU activation function introduces non-linearity into the model, enabling it to learn more complex patterns from the data. After extensive trails, the initial learning rate was determined to be 0.004. In this study, an adaptive learning rate strategy was employed to optimize the training process. Specifically, the learning rate was reduced to 80% of its current value whenever the model's loss failed to decrease over a span of 10 consecutive iterations. This approach ensures that the learning rate is dynamically adjusted to balance the speed of convergence and the stability of the training process, thereby improving the overall performance of the model. In our proposed method, the original data features are utilized in two parallel pathways: one involves feeding them into a neural network for forward propagation calculations, while the other involves directly inputting them into an empirical formula to predict the failure pressure. Before being fed into the neural network, the data features are normalized to a distribution with a mean of 0 and a standard deviation of 1, which helps to enhance the efficiency and accuracy of the network's learning process. In contrast, the data features input into the empirical formula are not subjected to any preprocessing, as the formula is designed to operate directly on the raw data. The loss function used in this paper is the mean squared error (MSE) loss function, and its calculation method is shown in Equation (8). \[Loss=\frac{\sum\limits_{i=1}^{n}\left(P_{F}-\hat{P}_{FA}\right)^{2}}{n} \tag{8}\]  
The prediction based on empirical formulas holds a certain degree of rationality due to their foundation in practical observations and historical data. However, these formulas often fall short when confronted with the variability and complexity of corrosion defects. Although the previous literature has successfully applied empirical formulas to guide the optimization of neural networks, thereby enhancing model performance [27; 28], directly applying empirical formulas to guide the optimization of neural networks, as illustrated in Equation (8), often proves to be counterproductive.