FG-PINNs: frequency-guided physics-informed neural networks for solving PDEs with high frequency components

TL;DR

The paper tackles the difficulty of learning high-frequency components in PDE solutions due to spectral bias by proposing FG-PINNs, a dual-subnetwork architecture that splits the solution into high- and low-frequency parts. The high-frequency subnetwork ingests a frequency prior derived from the problem data (e.g., the source term or initial/boundary conditions) and uses a frequency-aware forward rule, while the low-frequency subnetwork employs a standard fully connected network; the final solution is the sum $u_{nn}^{*} = u_{nn}^{h} + u_{nn}^{l}$. By reformulating the loss to incorporate frequency priors and balancing residuals with a term $ au = e^{-K_{ au}|f(oldsymbol{x})/f(oldsymbol{x})_{ ext{max}}|}$, FG-PINNs achieve rapid convergence and high accuracy across 1D Poisson, 1D heat, nonlinear and anisotropic 2D heat, and time-reversal PDE problems, with relative $L^{2}$ errors often in the $O(10^{-3})$–$O(10^{-4})$ range. The method reduces computational overhead compared to prior frequency-aware PINNs and demonstrates practical potential for solving PDEs with strongly oscillatory solutions, albeit with dependence on the availability of reliable high-frequency priors in $f$ or boundary/initial data.

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 core of the proposed algorithm lies in utilizing high-frequency information obtained from PDEs to guide the neural network in rapidly approximating the high-frequency components of the solution. 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 with high-frequency components into the neural network. 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 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: $O(10^{-3})$).

Figures (8)

• Figure 1: The neural network structure of FG-PINNs.
• Figure 2: Example \ref{['E1']}: Poisson equation (\ref{['e1']}). A: Exact solution and predicted solution. B: Absolute error. C: Exact spectrum and predicted spectrum. D: Convergence of the amplitude for each frequency during the training process.
• Figure 3: Example \ref{['E1']}: Poisson equation (\ref{['e1']}). A: $u_{nn}^{h}(x)$ spectrum. B: $u_{nn}^{l}(x)$ spectrum.
• Figure 4: Example \ref{['E2']}: Heat equation (\ref{['e2']}). A: Exact solution. B: Predicte solution. C: Absolute error. D: Relative $L^{2}$ error.
• Figure 5: Example \ref{['E3']}: Wave equation (\ref{['e3']}). A: Exact solution (t=0.5). B: Predicte solution (t=0.5). C: Absolute error. D: Relative $L^{2}$ error.

Theorems & Definitions (5)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5