Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems

A physics-informed neural network (PINN) combines the constraint-satisfaction ability of partial differential equations (PDEs) with the representation power of deep neural networks to learn solutions of PDEs. PINNs were first introduced in [3, 7, 11] as a way of solving problems in mathematical physics and engineering that can be modeled as PDEs. The idea behind PINNs is to treat the solution of a PDE as an unknown function that can be represented by a neural network. The neural network is then trained end-to-end to satisfy the boundary conditions and PDE constraints. This enables PINNs to deal with problems that are challenging to solve using conventional numerical techniques, such as, those with high-dimensional input spaces and complex boundary conditions. Due to the growing need for effective solutions to challenging physical problems in fields like fluid dynamics, structural mechanics, and heat transfer, PINNs have become increasingly popular in recent years. Computational and theoretical studies on PINNs have also shown to be useful for problems in machine learning, computer vision, and other fields outside physics and engineering due to their flexibility and representational power. PINNs have been applied to a variety of problems in physics, engineering, and other fields, including solving PDEs, modeling physical systems, and carrying out data-driven simulations. However, there are still some obstacles that arise when applying them to the field of computational science and engineering.