A PINN-based Approach for Solving the 1D Diffusion Equation in Stokes’ Second Problem

Abstract

In this study, a Physics-Informed Neural Network (PINN) is proposed for solving the one-dimensional diffusion equation under the oscillatory sine boundary condition. The PINN method differs from other numerical techniques that consume substantial computational power and require mesh construction since the physical equations and boundary conditions are integrated directly into the network architecture. This is accomplished through constructing a neural network via TensorFlow and employing a loss function that encompasses the physics equations and boundary conditions. The output of the PINN is then compared with the exact solution, showing remarkable consistency and numerical precision. The model has a consistent error of less than 0.02 and an RMSE of approximately 0.01, with absolute errors ranging from 10−3. Not only does the model demonstrate numerical superiority, but it can also accurately identify the phase and amplitude properties of the oscillatory solutions. In addition, it successfully transitions between areas of steep gradients at boundaries and regions of oscillatory damping to a steady state while maintaining temporal stability during training. Further, validation via topographic comparison of the three-dimensional model shows that the proposed numerical model retains the structure and physics of the analytical solution. In general, the results demonstrate the success of PINNs in providing an efficient and robust numerical approach for solving partial differential equations compared to classical and analytical approaches.