FG-PINNs: A neural network method for solving nonhomogeneous PDEs with high frequency components

Abstract: In this work, we propose the frequency-guided physics-informed neural networks (FG-PINNs), specifically designed for solving partial differential equations (PDEs) with high-frequency components. The proposed algorithm relies on prior knowledge about high-frequency components obtained from PDEs. It aims to use this prior knowledge to guide the neural network in rapidly approximating the solution's high-frequency components. The FG-PINNs consist of two subnetworks, including a high-frequency subnetwork for capturing high-frequency components and a low-frequency subnetwork for capturing low-frequency components. The key innovation in the high-frequency subnetworks is to embed prior knowledge for high-frequency components into the network structure. For nonhomogeneous PDEs ($f(x)\neq c, c\in R$), we embed the source term as an additional feature into the neural network. The source term typically contains partial or complete frequency information of the target solution, particularly its high-frequency components. This provides prior knowledge about the high-frequency components of the target solution. For homogeneous PDEs, we embed the initial/boundary conditions with high-frequency components into the neural network. Based on spectral bias, we use a fully connected neural network as the low-frequency subnetwork to capture low-frequency components of the target solution. A series of numerical examples demonstrate the effectiveness of the FG-PINNs, including the one-dimensional heat equation (relative $L^{2}$ error: $O(10^{-4})$), the nonlinear wave equations (relative $L^{2}$ error: $O(10^{-4})$) and the two-dimensional heat equation (relative $L^{2}$ error: $O(10^{-3})$).