## 1 Introduction  
In three-dimensional (3D) space, transformations can be classified as rigid body and non-rigid transformations. Rigid body transformation is a special case of a similar transformation that connects point sets in different coordinate systems through rotation and translation [1]. Non-rigid transformation allows translation, rotation, scaling, and more complex deformation operations on an object, through which the position, direction, size, and shape of the object in a 3D space can be changed. Compared with rigid transformation, non-rigid transformation usually requires more complex mathematical models and algorithms; however, the advancement of modern computer and graphics processing technology enables real-time performance of non-rigid transformation. This has significant implications for areas such as virtual reality, augmented reality, and interactive applications. Three-dimensional deformation is a central research topic in the field of flexible transformation. [PERSON] et al. [2] monitored the physical deformation of large structures by extending one-dimensional (1D) two-level piezoresistive behavior to 2D/3D surface self-sensing. [PERSON] et al. [3] determined the material parameters of [PERSON]'s third-order model based on a uniaxial test. The authors established a mathematical model of the input pressure and deformation of a water hydraulic soft actuator. [PERSON] et al. [4] developed a small-scale soil deformation measurement system based on a fiber Bragg grating for underground displacement monitoring. [PERSON] et al. [5] proposed a method to estimate the deformation and strain of an entire structure by measuring the discrete 3D coordinate data of the target structure obtained from LiDAR. [PERSON] et al. [6] implemented digital image correlation technology to accurately measure the full-field strain generated in various quasi-static tensile tests, including cyclic tensile testing, uniaxial tensile testing, and stress relaxation tests, which accurately measured the characteristics of polydimethylsiloxane that is directly related to the performance of wearable strain sensors. Note that research on deformation monitoring based on FEM methods typically involves a complex metamodeling process that is time consuming and labor intensive. With the advent of Industry 4.0, the need for 3D deformation intelligent detection and digital twins is increasing. Mesh geometry can adapt to various domains as an input to artificial neural networks [7]. Therefore, some scholars have realized 3D non-rigid transformation deduction by introducing deep-learning frameworks. For instance, [PERSON] et al. [8] designed a learning method based on a deep neural network to correct constant curvature kinematics through accurate visual estimation results. [PERSON] et al. [9] used artificial neural network (ANN) technology to simultaneously measure temperature and strain to integrate fiber Bragg gratings. [PERSON] et al. [10] reported an AI-based method, which implemented a generation confrontation neural network based on game theory, to bridge the gap between the material microstructure (design space) and physical properties. [PERSON] et al. [11] predicted the strain and stress tensor for a given input composite geometry based on deep learning methods. [PERSON] et al. [12] proposed a new DIC method based on deep learning Deep DIC, where two convolutional neural networks (CNNs), Displacement-Net and StrainNet, collaborate to predict the displacement and strain end-to-end. Note that the 3D non-rigid transformation deduction based on the deep learning method can learn robustness from data while still performing well in the case of limited noise. However, after the deep-learning model is trained, the 3D non-rigid transformation result can be deduced from the new input data with a low delay. The contributions of this paper are as follows:  
* We propose a 3D non-rigid transformation derivation model based on multi-control point perception data to predict the shapes of entities after transformation.