Abstract Summary
Moving load problems are ubiquitous in engineering, where an engineering structure is subjected to a moving point mass source, for instance, a vehicle moving on a bridge, a train moving on railway tracks, and a pantograph moving on a catenary, among others. The moving point source induces forces on the structure, which subsequently causes the deformation and displacement of structures. One physical quantity of interest for engineers is the deflection of the structure, determining which benefits the design and maintenance of the structure. In practice, numerical studies are carried out to compute the deflection profiles. However, in the case of railways, where the tracks are hundreds of kilometres long and the deflection of the tracks is in the scale of micro and nanometers, obtaining the complete deflection profiles of the railway tracks and catenaries is challenging. Mathematically, moving loads on beam like structures can be modelled as Partial Differential Equations (PDEs), including a Dirac delta function modelling the moving point mass source. These PDEs could be simulated to obtain complete deflection profiles at all space time locations. In recent years, considerable interest has shifted towards employing deep learning based methods to simulate and model differential equations. Traditionally deep learning based methods require a substantial amount of data; however, collecting the data for deflection profiles in case of moving loads is experimentally expensive. One possible way to mitigate this challenge is to leverage the physical equations directly in the neural network's loss function. Recently, a deep neural network based architecture termed as Physics Informed Neural Networks (PINNs) has been proposed to solve physical equations that use the PDEs in the loss function along with initial and boundary conditions and does not need any experimental data. Although PINN has shown significant progress in solving systems of physical equations, including non linear engineering systems, there are also several limitations of PINNs, including its inability to tackle shock type behaviour. The Dirac delta function in moving load PDE models an instantaneous load change at one location, which poses a challenge in the PINNs training process. We aim to tackle this challenge and accurately simulate the moving load problem through vanilla PINN architecture. In this work, we approximate the Dirac delta function as a Gaussian distribution. This results in a smooth approximation of the Dirac delta function, mitigating the challenge for the vanilla PINN architecture. In this paper, numerical experiments will be presented, highlighting the successful implication of the proposed methodology for moving load problems having applications in railway engineering. The major outcome of this research will be 1.) To the author's best knowledge this is the first work which simulates the moving load problems through Physics informed machine learning. 2.) Several experiments will be presented for the train track and catenary pantograph interactions, establishing the proposed methodology's validity. 3.) From the perspective of machine learning, this work would propose an approach to tackle the Dirac delta function efficiently.
