arXiv:2603. 24002v3 Announce Type: replace Abstract: Physics-Informed Neural Networks (PINNs) for high-dimensional and high-order partial differential equations (PDEs) are primarily constrained by the $\mathcal{O}(d^k)$ spatial derivative complexity and the $\mathcal{O}(P)$ memory overhead of backpropagation (BP).
By Zhangyong Liang, Huanhuan Gao
arXiv:2604. 09361v3 Announce Type: replace Abstract: This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain.
By Zhangyong Liang, Tingfeng Wang, Xiaofei Zhao
arXiv:2607. 20378v1 Announce Type: new Abstract: Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability.
By Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee, Aida Pakniyat, Dara Rahmati, Saeid Gorgin
arXiv:2606. 19912v1 Announce Type: cross Abstract: We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system.
By Yunlong Li, Fei Wang
The paper introduces Spectrally Optimised Neural Discretisations (SpeND), a mesh‑free framework that learns discretisation weights from local stencil geometry on unstructured point clouds. By embedding discrete moment conditions into the network architecture, SpeND guarantees polynomial consistency and allows the weights to be optimised for spectral accuracy over a chosen wavenumber band, using an unsupervised Fourier‑mode loss. The resulting operators are PDE‑agnostic, perform well on Poisson, Burgers, and Navier–Stokes equations, and can reduce wall‑clock time by 3–20× compared to existing mesh‑free methods at the same accuracy.
By Lucas Gerken Starepravo, Henry Broadley, Steven Lind, Jack R. C. King
arXiv:2607. 22004v1 Announce Type: new Abstract: Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain.
By Zhangyong Liang, Huanhuan Gao