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:2605. 31027v2 Announce Type: replace Abstract: We propose a novel neural network architecture, termed Multi-Scale Separable Fourier Neural Networks (MS-SFNN), for the accurate and efficient solution of linear and nonlinear high-frequency partial differential equations (PDEs).
arXiv:2608. 05892v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems.
arXiv:2607. 13566v1 Announce Type: cross Abstract: For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy.
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.
Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions. We show these failures are multi-causal, arising from the concurrent interplay of (i) spectral bias against sharp features, (ii) imbalanced multi-term optimization and loss-weight collapse, (iii) violation of temporal causality, and (iv) under-resolved collocation.
arXiv:2602. 06842v2 Announce Type: replace-cross Abstract: Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate convergence.
arXiv:2510. 19399v2 Announce Type: replace Abstract: Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs).
Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
The paper introduces the Frequency Selective Neural Network (FSNN), a new foundation architecture for time‑series learning that embeds advanced signal‑processing mathematics into its neural topology. By using a fully differentiable Wiener‑like filter bank optimized with complex‑domain backpropagation, FSNN autonomously discovers and isolates the precise physical modes of a given task, thereby avoiding the spectral entanglement that plagues CNNs, RNNs, and Transformers. Extensive evaluations show that FSNN achieves state‑of‑the‑art predictive performance, attaining 77.0 % average accuracy on the 10 multivariate UEA datasets and leading all major metrics on the imbalanced PTB‑XL ECG benchmark, while converging directly on physically meaningful frequency bands such as the cardiac QRS complex.
arXiv:2606. 16575v2 Announce Type: replace 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:2607. 11310v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions.