## 1 Introduction
The physical rules are learned by adding physical equations, boundary conditions, and initial conditions as penalty terms to the loss function during the training process. These penalty terms penalize solutions that do not satisfy the physical conditions, ensuring that the training results meet the physical laws. Compared to purely data-driven neural networks, the PINN can achieve better training results with less training data. The PINN does not require grid generation or discretization for solving problems and is not affected by the computational step size. This approach is highly suitable for solving ordinary differential equations (ODEs) and PDEs. Therefore, the possibility of adopting the PINN to solve differential equations has garnered increasing attention [23]. [PERSON] et al. [24] proposed a self-learning method based on deep reinforcement learning to solve nonlinear differential equations and partial differential equations. Using this method, they successfully obtained accurate solutions for equations such as Burgers and Lorenz. [PERSON] et al. [25] presented the application of PINNs in industrial heat transfer problems by analyzing two prototype problems of forced and mixed convection. [PERSON] et al. [26] introduced a parallel physics-informed neural network algorithm that significantly improves the computational efficiency of the long-term integration of partial differential equations. [PERSON] et al. [27] developed a Bayesian physics-informed neural network for solving PDEs and nonlinear problems described by noisy data, effectively quantifying uncertainty and avoiding overfitting. [PERSON] et al. [28] proposed the PINN-Stress model, which combines a PINN with finite element simulation to predict the stress distribution in gusset plates under dynamic loads. This study employs a fully connected neural network as a function approximator to solve the motion equations of structural systems and obtain displacement response outputs. The loss function comprises a combination of motion equations, boundary conditions, and initial conditions as loss terms. The PINN is constructed based on the adaptive moment estimation (Adam) optimization algorithm. Numerical studies were conducted on structural systems with two degrees of freedom, four degrees of freedom, and a cantilever beam under various loading conditions. These examples verify the accuracy and efficiency of the PINN model proposed in this paper for solving differential equations. The remaining sections of this paper are structured as follows: In Section 2, the composition structure of the PINN is introduced, and detailed descriptions of the embedding method for physical information, the setup of the loss function, and the training rules are provided. The third section uses the proposed method to solve the motion equations of three damped multiple-degree-of-freedom systems under forced vibrations. The displacement results for each degree of freedom are computed using this approach. The accuracy of the predictions is validated by comparing them with the results obtained using the Newmark-\(\beta\) method. Finally, the conclusions of this paper are presented in Section 4.