Frequency Decomposition Benefits Physics-Informed Neural Networks for Complex PDEs

Physics-Informed Neural Networks (PINNs) are a powerful tool for solving partial differential equations (PDEs) by embedding governing equations into the training process. However, PINNs often struggle with high-frequency and multi-scale features due to spectral bias, where they learn low-frequency components more readily. Techniques like Fourier feature embeddings and sinusoidal activations aim to mitigate this, but their universal benefit has been largely assumed. This preliminary study introduces a dual-branch, spectrally-gated architecture (DBSG-PINN) to systematically investigate the impact of frequency decomposition and spectral routing. This architecture splits low and high-frequency components into separate subnetworks, adaptively joined by a gate. Testing on five one-dimensional PDEs, the research found that frequency decomposition significantly helps with spectrally complex, multi-scale problems, reducing relative L2 error by up to 59.2% on a multimodal wave problem. Conversely, it offered minimal or even negative benefit for smoother PDEs. The adaptive gate's effectiveness correlated with the spectral richness of the solution, suggesting it exploits frequency structure. This exploratory study highlights the need for further investigation with more seeds and benchmarks.