arXiv:2607. 13566v1 Announce Type: cross Abstract: For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy.
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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:2602. 09708v2 Announce Type: replace-cross Abstract: We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems.
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:2609.07983v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving Partial Differential Equations (PDEs), offering a me...
arXiv:2607. 06348v1 Announce Type: new Abstract: We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations.
The paper introduces Spectral-like Neural Discretisation (SpeND), a mesh‑free method that learns stencil weights via a neural network to approximate the modal response of a spectral operator across a specified band of wavenumbers. By projecting the network output onto the space of polynomial‑consistent weights, SpeND ensures exact consistency while minimizing dispersion and dissipation errors in a self‑supervised, physics‑agnostic manner. Experiments on disordered 2‑D node sets demonstrate that the learned fourth‑order operator matches the exact spectral response over a wider band than traditional LABFM or structured‑grid finite differences, and retains fourth‑order convergence upon refinement.