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

Physics-Informed Neural Networks (PINNs) have shown great potential for solving partial differential equations (PDEs). However, their performance is often limited when dealing with multi-scale structures and high-frequency solution components due to spectral bias and training instability. To address these challenges, a novel framework called the Fourier Feature-Enhanced Multi-layer Residual Stacking Network (FRES) is proposed. The key idea of FRES is to leverage residual information to guide frequency-domain modeling, thereby enabling more accurate representation of complex multi-scale features. Specifically, FRES employs a dynamic Fourier embedding mechanism to adaptively generate high-frequency features, while a multi-layer residual correction scheme progressively reduces prediction errors. In addition, a frequency-enhanced residual fusion module integrates low-frequency structures with high-frequency details, and a trainable frequency matrix further improves adaptability across different problems. We systematically evaluate FRES on some representative PDE benchmarks, including the Burgers, Schrödinger, and KdV equations. Experimental results demonstrate that FRES significantly outperforms traditional PINNs and multi-scale Fourier feature methods (MFF) in terms of accuracy, extrapolation capability, and noise robustness, highlighting its potential for robust and efficient modeling of complex physical fields.