## 1 Introduction  
The dynamic response calculation of a foundation beam under moving loads represents a highly complex mechanical issue that integrates knowledge and techniques from multiple disciplines. Addressing this problem not only deepens our understanding of dynamic responses in practical engineering scenarios but also offers theoretical and technical support to ensure structural safety and reliability [1, 2]. The solutions to this problem typically involve analytical and numerical methods. Analytical methods necessitate the formulation of mathematical models, followed by the application of integral transforms (e.g., Laplace, Fourier) to solve the governing equations. This process yields closed-form solutions that offer clear insights into how different parameters influence dynamic responses [3, 4]. However, these methods are generally restricted to idealized models and specific loading conditions. In contrast, numerical methods, which often utilize the finite element method (FEM), are capable of tackling complex real-world engineering problems. They can manage irregular geometries, complex boundary conditions, and multiphysics couplings but demand significant computational resources [5]. Furthermore, a single finite element simulation can only provide the solution for a particular case. When variables such as material properties, structural models, or loading patterns change, remodeling and remeshing become necessary. This lack of adaptability renders finite element simulation inefficient for certain problems [6]. In recent years, deep learning (DL) techniques, celebrated for their remarkable fitting capabilities, have garnered significant attention in the analysis and prediction of dynamic responses induced by moving loads [7; 8; 9]. The effectiveness of these DL methods largely hinges on supervised training, where the model adjusts its parameters by calculating the error between predicted and true labels (loss function) and then uses backpropagation and optimizers (e.g., gradient descent) to minimize this error. Repeated iterations of this process enable the model to improve and achieve high accuracy on test data [10; 11]. However, in practical applications, obtaining high-quality monitoring data is often fraught with difficulties [12]. As a result, accurately predicting the dynamic response of structures becomes extremely challenging when data are scarce or unavailable. Recently, physics-informed neural networks (PINNs), which combine scientific computing and machine learning, have shown considerable potential for solving partial differential equations (PDEs) in data-scar scenarios [13; 14; 15]. Research on the application of PINNs to beam bending problems, particularly dynamic responses under moving loads, is still in its early stages. [PERSON] et al. [16] explored single--and double--beam structures using PINNs based on Euler-Bernoulli (E-B) and Timoshenko theories. Their work proved PINNs' effectiveness in solving forward and inverse problems in complex beam systems. In follow-up studies, by modeling moving loads with Gaussian functions, PINNs were used to calculate beam deflection and predict moving loads [17]. However, this method treated the moving load as a static load progressing across different positions, neglecting dynamic effects. [PERSON] and [PERSON] [18] applied PINNs to study the bending, natural frequencies, and free vibrations of 3D functionally graded porous beams, analyzing how material distribution, elastic foundation, porosity factor, and porosity distribution type affect bending response and natural frequencies. However, their research focused only on the spatial conditions under external loads, ignoring complex spatiotemporal effects. [PERSON] et al. [19] approximated the Dirac function with a Gaussian function, introduced a Fourier embedding layer into PINNs' deep neural network, and incorporated causal weights into the loss function to obtain beam dynamic responses under moving loads.