## 1 Introduction  
With the rapid development of industries such as aerospace and automotive manufacturing, in order to cope with the energy and environmental crisis, structural lightweighting has also become one of the keys to the healthy and sustainable development of the aerospace and automotive industries nowadays. The main way to achieve lightweighting is through the lightweighting of materials and structures [1]. As a representative of hollow parts, large cross-section differential variable diameter tubes are widely used in the exhaust hydraulic systems of automobiles, rockets, or aircraft, as well as in load-bearing structural components [2]. These parts are traditionally manufactured by stamping out several pieces and then welding them together. This method seriously affects the reliability and service life of the variable diameter tube as the subsequent assembly is difficult, and the process steps are numerous. Tube hydroforming technology is a process in which high-pressureliquid is applied on the inside the tube, causing the tube to expand and form complex shapes under the constraint of the die. This process can reduce the weight of parts, shorten the production cycle, and also decrease the material waste caused by subsequent welding. The use of flexible media to transmit force breaks through the limitations of the shape of the core punch and can be formed at once according to the target part with multiple variations in diameter or complex cross-sectional shapes. It is now widely used in the manufacture of hollow parts for automobiles. The study of the hydroforming process for the variable diameter tube mainly lies in the forming process, and the process parameters must be reasonably designed; otherwise, it is prone to defects such as buckling, wrinkling, and rupture. Rupture is due to local instability caused by excessive tensile stresses during the expansion process. Buckling and wrinkling are due to compression instability caused by excessive axial force or low internal pressure [3]. In order to reduce the appearance of defects in the production process, many scholars have concentrated on the matching relationship between internal pressure and axial feeds. Since the hydroforming process of tubes is a dynamic process of change, it is not possible to accurately express the process between the forming parameters by an accurate mathematical model or a simple quantitative functional relationship. This bottleneck that we face places demands on us to break away from traditional methods and find new approaches. Machine learning is an important method in the field of artificial intelligence. The back propagation neural network (back propagation, BP) is a type of machine learning algorithm, the core of which is based on the error back propagation algorithm. The weights and thresholds are continuously adjusted through algorithms such as the gradient descent method according to the least squares idea, so that the difference between the final desired value and the actual value is minimised [4]. Commonly used optimisation algorithms are the particle swarm algorithm, genetic algorithm, grey wolf algorithm, annealing algorithm, etc. [PERSON] et al. [5] firstly applied the fuzzy control algorithm to the optimisation of the loading path, and, compared with the traditional optimisation method, the thinning rate of the tube after optimisation using the fuzzy control was significantly reduced, which reduced the probability of rupture during the tube deformation process. [PERSON] et al. [6] conducted a study with pressure, internal pressure, and friction as input parameters. A fast NSGA- \(\mid\mid\) algorithm based on the artificial neural network (ANN) with solving multi-objectives determines the optimal combination of hydroforming parameters to satisfy the filling rate and the thinning rate of the foot of a double-stepped tube. Combined with finite element (FE) simulations, results were obtained showing that the feed rate has a significant effect on thinning rates and corner filling. [PERSON] et al.