arXiv:2610. 21964v1 Announce Type: cross Abstract: Neural networks often learn low-frequency components before high-frequency ones --- a phenomenon called spectral bias or termed F-principle.
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The paper investigates when frequency decomposition aids Physics-Informed Neural Networks (PINNs) by introducing a dual‑branch, spectrally‑gated architecture (DBSG‑PINN) that separates low‑ and high‑frequency components. Experiments on five one‑dimensional PDE benchmarks show that frequency decomposition significantly reduces error—up to 59.2% on a multimodal wave problem—when the target solution is spectrally complex, but offers little improvement on smoother problems and can even worsen performance on a simple 1D wave benchmark. The adaptive gate’s effectiveness scales with the spectral richness of the solution, suggesting it exploits frequency structure rather than adding noise.
arXiv:2606. 16575v1 Announce Type: new Abstract: Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors.
arXiv:2402.17595v2 Announce Type: replace-cross Abstract: The phenomenon of implicit bias has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neur...
arXiv:2609.15355v1 Announce Type: cross Abstract: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural network...
The paper analyzes gradient flow in a two‑layer linear neural network as the input, hidden, and output dimensions grow proportionally. Using a closed‑form solution combined with random matrix theory, it derives exact asymptotic spectral distributions for the weight matrices at all times and tracks how the learned subspace evolves. In the limit of small initialization, the study reveals a silent alignment phase where singular directions align quickly while singular values and loss stay unchanged, followed by a visible learning phase on a logarithmic timescale where singular values increase and loss decreases through a saddle‑to‑saddle dynamic that becomes continuous in the high‑dimensional limit.
arXiv:2606. 28486v1 Announce Type: cross Abstract: The emergence of low-dimensional structures in the spectra of neural network weight matrices is a common empirical feature of trained models, but the dynamical origin of this phenomenon during learning remains an open problem.