## 1 Introduction  
Gas-liquid two-phase flow is a common flow across various industries, including petroleum, chemical, natural gas, thermal power generation, aerospace, and nuclear energy. Accurately measuring parameters in such flows is crucial for the efficient and stable operation of industrial systems and production activities. However, due to the complexity of affecting factors like flow structure, physical characteristics, flow rate, pressure, heat treatment, pipeline, and geometry, it is an enormous challenge to perform accurate measurements using traditional methods [1]. Currently, the measurement of gas-liquid two-phase flow rates in gas, oil, and chemical processes often involves the use of separation methods, which require a large space and expensive equipment [2]. While this challenge can be effectively resolved using non-separation measurement methods, they have limitations in terms of measurement range and medium. For instance, the performance of Doppler technology-based methods is affected by flow patterns, which may adversely affect the measurement accuracy. Additionally, there are significant differences in physical properties such as density and sound speed between gases and liquids, which may affect ultrasound propagation and measurement accuracy [3]. Therefore, it is of significant importance to develop a low-cost, accurate, and convenient method for measuring gas-liquid two-phase flow rates. Studies show that intelligent learning algorithms have significant advantages in addressing nonlinear and multivariate problems and can effectively improve the applicability of non-separation measurement methods. Consequently, exploring the application of intelligent learning algorithms in measuring gas-liquid two-phase flow rates has become a hot research topic in recent years. In this context, [PERSON] et al. [4] utilized acoustic emission (AE) and near-infrared (NIR) sensors to capture resonance peak edge factors, and subsequently employed the least absolute shrinkage selection operator (LASSO) machine learning technique to predict gas-liquid two-phase void fraction and flow rate. This approach achieved an 89.47% accuracy within a 20% relative error margin for emulsion flow data. [PERSON] et al. [5] proposed a physics-guided deep learning model by integrating a physical constraint term of the rates of change of pressure caused by the slip term to maintain consistency in the gas and liquid phases into the loss function of a deep neural network. This physics-guided neural network (PGNN) successfully measured gas and liquid flow rates simultaneously with a high accuracy, even under a wide range of test conditions (gas volume fraction (GVF) > 95%). Furthermore, [PERSON] et al. [6] predicted gas and liquid flow rates in intermittent flows using information from a cone gauge and a conductivity loop sensor. The predicted liquid mass flow rate had a relative error of less than \(\pm\)5.0% at a 99% confidence level, while for the gas phase, the relative error was within \(\pm\)15.0% at a 98% confidence level. [PERSON] et al. [7] applied convolutional neural networks (CNNs) to multiphase flow rate prediction, introducing the flow adversarial network (FAN). Experimental results demonstrated a mean absolute percentage error (MAPE) of up to 8.36% and 9.09%. [PERSON] et al. [8; 9] developed a multi-task-based time-channel convolutional neural network (MTCCNN) to predict porosity, achieving a superior performance in gas phase gas content prediction. Additionally, they designed a segmented dense connectivity network (SDCN), capable of simultaneously measuring the total rate and void ratio using signals from a four-sector conductivity sensor. The framework exhibited a high accuracy in the total flow measurement of gas-liquid two-phase flow within a range of 0.18-1.23 m\({}^{3}\)/h. Integrating sensors with intelligent learning models is a complex process with high measurement requirements and associated costs.