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.
By Yulun Wu, Matthieu Barreau, Miguel Aguiar, Karl H. Johansson
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:2502. 07209v4 Announce Type: replace Abstract: Physics-Informed Neural Networks (PINNs) seek to solve partial differential equations (PDEs) with deep learning.
By Shaghayegh Fazliani, Zachary Frangella, Madeleine Udell
arXiv:2608. 14733v1 Announce Type: cross Abstract: Building on the foundation of single-hidden-layer neural networks, Fourier Feature Networks (FENs) are proposed, which incorporate Fourier features using $\cos$, $\sin$, or a combination of both.
By Qihong Yang, Zhijie Su, Yangtao Deng, Qiaolin He
arXiv:2607. 02199v1 Announce Type: cross Abstract: Mutual information (MI)-inspired feature learning techniques are capable of generating low-dimensional embeddings that retain nonlinear dependence structures, but direct estimations of MI suffer from noisy probability distribution estimates in the low-data regime.
By Preston Pitzer, Anish Pradhan, Harpreet S. Dhillon
arXiv:2607. 13566v1 Announce Type: cross Abstract: For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy.
By Tianchi Yu, Ivan Oseledets