Fourier feature-enhanced multi-layer residual stacking network: A novel multiscale modeling approach for...

Area of Science:

• Computational physics
• Machine learning for differential equations

Background:

• Physics-Informed Neural Networks (PINNs) show promise for solving partial differential equations (PDEs).
• PINNs face limitations with multi-scale structures and high-frequency components due to spectral bias and training instability.
• Existing methods often lack robustness and accuracy for complex physical phenomena.

Purpose of the Study:

• Introduce a novel framework, Fourier Feature-Enhanced Multi-layer Residual Stacking Network (FRES), to overcome PINN limitations.
• Enhance the representation of complex multi-scale features and high-frequency solution components.
• Improve accuracy, extrapolation capability, and noise robustness in solving PDEs.

Main Methods:

• FRES utilizes residual information to guide frequency-domain modeling.
• A dynamic Fourier embedding mechanism adaptively generates high-frequency features.
• A multi-layer residual correction scheme and frequency-enhanced residual fusion module progressively reduce errors and integrate features.
• A trainable frequency matrix enhances adaptability.

Main Results:

• FRES demonstrates superior performance on benchmark PDEs (Burgers, Schrödinger, KdV equations).
• Significantly outperforms traditional PINNs and multi-scale Fourier feature methods (MFF).
• Exhibits enhanced accuracy, extrapolation capabilities, and noise robustness.

Conclusions:

• FRES offers a robust and efficient solution for modeling complex physical fields governed by PDEs.
• The proposed framework effectively addresses spectral bias and training instability in PINNs.
• FRES shows significant potential for advancing scientific machine learning applications.