Densely Multiplied Physics Informed Neural Networks

Abstract: Although physics-informed neural networks (PINNs) have shown great potential in dealing with nonlinear partial differential equations (PDEs), it is common that PINNs will suffer from the problem of insufficient precision or obtaining incorrect outcomes. Unlike most of the existing solutions trying to enhance the ability of PINN by optimizing the training process, this paper improved the neural network architecture to improve the performance of PINN. We propose a densely multiply PINN (DM-PINN) architecture, which multiplies the output of a hidden layer with the outputs of all the behind hidden layers. Without introducing more trainable parameters, this effective mechanism can significantly improve the accuracy of PINNs. The proposed architecture is evaluated on four benchmark examples (Allan-Cahn equation, Helmholtz equation, Burgers equation and 1D convection equation). Comparisons between the proposed architecture and different PINN structures demonstrate the superior performance of the DM-PINN in both accuracy and efficiency.