## 1 Introduction  
In engineering, various complex problems are often represented and described using differential equations. Among them, the equations of motion are widely applied to describe and analyze the dynamic behavior of structural systems, including vibrations, displacements, and accelerations [1]. These differential equations provide crucial information about structural response and behavior, assisting engineers in designing safe, stable, and reliable civil structures. While analytical methods can yield exact solutions to differential equations, they have limitations such as a high computational workload and strict applicability conditions. Moreover, mathematical models based on physical laws may not always have analytical solutions, making it challenging to resolve most practical problems [2]. Therefore, numerical methods are commonly employed for approximate solutions. Popular analytical methods include the finite element method [3], finite difference method [4], boundary element method, and Newmark-\(\beta\) method [5]. Although numerical methods for solving differential equations have made significant progress and can solve many engineering and physical problems, these methods still face challenges such as high computational complexity, nonconvergence, large errors, and error accumulation. In the past few decades, machine learning techniques, represented by neural networks, have achieved remarkable advancements and provided new approaches to problem solving in various fields [6; 7; 8; 9]. Many researchers have utilized artificial neural networks (ANNs)to develop different methods [10] for solving differential equations. [PERSON] et al. [11] proposed a general method for solving differential equations using artificial neural networks as function approximators. [PERSON] et al. [12] introduced a novel constrained integration method for solving partial differential equations by combining classical Galerkin methods with artificial neural networks. [PERSON] et al. [13] utilized artificial neural networks to calculate tunneling profiles for cosmological phase transitions, obtaining approximate or more exact results than dedicated differential equation solvers. [PERSON] et al. [14] presented a method for solving partial differential equations using deep feedforward neural networks. This method was applied to solve examples of advection- and diffusion-type partial differential equations (PDEs) in 1D and 2D, demonstrating superior performance over traditional grid-based methods. However, classical machine learning algorithms train models using known data to establish the mapping relationship between the input and output data, training a specific neural network model. Furthermore, purely data-driven approaches may exhibit an excellent fit to observations, but the predicted results could be physically inconsistent or implausible [15; 16]. This is mainly due to the challenges posed by extrapolation and observational biases, which can result in limited generalization performance. Moreover, in practical engineering applications, training data often satisfy certain physical rules. Purely data-driven neural network algorithms cannot consider this valuable information, wasting information resources or limiting their application in practical engineering scenarios. In recent years, with the explosive growth of data and computing resources, computer technology has experienced rapid development, and machine learning techniques, such as image recognition and language processing, have achieved significant advancements in various fields [8; 17; 18; 19; 20; 21]. The rapid development of machine learning techniques has also brought new opportunities for solving differential equations. [PERSON] et al. [22] proposed the PINN in 2017. The PINN is a type of neural network used to solve supervised learning tasks. It can learn the mapping relationship between the input and output in training samples. Additionally, the physical laws described by mathematical physics equations can be learned.