Constrained Self-Adaptive Physics-Informed Neural Networks with ResNet Block-Enhanced Network Architecture

Physics-informed neural networks (PINNs) have been widely adopted to solve partial differential equations (PDEs), which could be used to simulate physical systems. However, the accuracy of PINNs does not meet the needs of the industry, and severely degrades, especially when the PDE solution has sharp transitions. In this paper, we propose a ResNet block-enhanced network architecture to better capture the transition. Meanwhile, a constrained self-adaptive PINN (cSPINN) scheme is developed to move PINN's objective to the areas of the physical domain, which are difficult to learn. To demonstrate the performance of our method, we present the results of numerical experiments on the Allen-Cahn equation, the Burgers equation, and the Helmholtz equation. We also show the results of solving the Poisson equation using cSPINNs on different geometries to show the strong geometric adaptivity of cSPINNs. Finally, we provide the performance of cSPINNs on a high-dimensional Poisson equation to further demonstrate the ability of our method.