## 1 Introduction  
Pipelines are economical and safe means for transporting and distributing large volumes of oil and gas products across great distances [1]. However, pipeline failures can lead to severe consequences such as fatalities, economic losses, and environmental damages [2]. Among the various failure mechanisms threatening the structural integrity of pipelines, corrosion is a leading threat [3]. The in-line inspection (ILI) tool is widely used for detecting and measuring corrosions to assess pipeline integrity [4], and magnetic flux leakage (MFL) is the predominant ILI technology for identifying internal and external corrosion defects in both liquid and gas pipelines [5]. The Pipeline Operators Forum (POF) suggests that a single corrosion anomaly on the internal or external surface of a pipeline can be categorized into one of seven classes based on the ILI-reported corrosion length and width as illustrated in Figure 1[6], i.e., axial slotting, circumferential slotting, axial growing, circumferential grooving, general, pinhole and pitting, to better quantify the accuracy of ILI tools and facilitate the selection of appropriate models to evaluate the burst capacity of pipelines containing corrosion defects [6; 7]. The estimation of the corrosion anomaly sizes based on MFL signals falls into the inverse modeling realm. The pipeline industry frequently employs the iterative inverse model because of its remarkable precision [8; 9; 10]. However, the iterative inverse model is computationally costly. Inverse models based on machine learning provide promising alternatives to sizing corrosion defects from MFL signals with high accuracy but are more computationally efficient compared to the iterative inverse models [11; 12; 13]. For instance,Kandroodi et al. [14] fed the algorithm-estimated defect width and length, as well as the extracted peak-to-peak values from the MFL signals, into a Gaussian radial basis function neural network to predict the corrosion depth. [PERSON] et al. [15] proposed an error adjustment methodology for reconstructing the corrosion profiles from MFL signals based on a radial basis function neural network. However, these models have certain limitations: the training of these models relies on manually chosen features from the MFL signals, potentially omitting valuable data necessary for accurately predicting the corrosion profile. Additionally, the majority of these techniques focus solely on one MFL signal dimension, neglecting the critical information present in the other two dimensions. Over the past decade, convolutional neural networks (CNNs) have risen as a revolutionary deep learning algorithm, yielding remarkable results across various pattern recognition fields, including image and voice processing [16; 17]. The CNN tool can automatically extract deep features from input data in place of manually designed feature extractors; therefore, it is well suited to solve the inverse modeling based on the MFL signals. The application of CNN in the inverse analysis of MFL signals has been reported in the literature. [PERSON] et al. [18] introduced the visual transformation CNN, which predicts the length, width, and depth of anomalies by taking into account two components (i.e., circumferential and radial) of the MFL signal while disregarding the longitudinal component. [PERSON] and [PERSON] [19] employed CNN to estimate the locations and dimensions of corroisons on steel pipelines based on three components of the MFL signals. [PERSON] and [PERSON] [20] proposed a CNN architecture comprising two main modules: one for anomaly classification and another for anomaly size regression. The input of the classification module consists of all three components of the MFL signal, and the outputs are seven anomaly categories, as illustrated in Figure 1. The regression module incorporates seven separate CNNs, each tailored to a specific anomaly type, predicting dimensions such as width, length, and depth of the defect.