The paper introduces SFRBE–RaNN, a two‑stage adaptive randomized neural network framework for solving partial differential equations. It first redistributes a fixed feature budget based on regional indicator masses and optimizes regional weight scales using low‑dimensional residuals, then enriches the trial space by selecting nonredundant activation families through residual verification. The authors provide theoretical guarantees for linear problems, including a discrete residual‑reduction identity, error bounds, and a probabilistic success bound, and validate the approach with numerical experiments on localized, multiscale, and nonlinear space–time PDEs.
By You Yang, Fei Wang
The paper investigates fixed‑design random feature ridge regression beyond the mean‑kernel approximation, focusing on how the predictor’s nonlinear dependence on the empirical kernel affects training error, test error, and the conditional generalization gap. By deriving covariance‑level (one‑loop) corrections via a finite resolvent identity, the authors avoid an almost‑sure Neumann‑series assumption and provide explicit remainder bounds for training error. Numerical experiments demonstrate that including mixed train–test covariance tensors is essential for accurate test error predictions and reveal a width–regularization boundary where second‑order truncation becomes unreliable.
By Taeyoung Kim
arXiv:2609. 16406v1 Announce Type: cross Abstract: Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years.
By Chi-An Chen, Chunyang Liao, Ming Zhong
arXiv:2607. 10546v1 Announce Type: new Abstract: Discovering governing partial differential equations (PDEs) from noisy observational data is a fundamental challenge in scientific machine learning.
By Jinyang Du, Hao Ma, Xiaohu Shi, Bo Yang, Yanchun Liang, Heow Pueh Lee, Chunguo Wu
arXiv:2606. 29440v1 Announce Type: new Abstract: Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training.
By Zirui Deng, Jingbo Sun, Deyu Meng, Fei Wang
arXiv:2605. 26631v2 Announce Type: replace-cross Abstract: We propose KO-PDE-IDENT, a data-driven framework for identifying parsimonious partial differential equations (PDEs) with false discovery rate (FDR) control.
By Pongpisit Thanasutives, Naichang Ke, Yoshinobu Kawahara
arXiv:2610.26458v2 Announce Type: replace-cross
Abstract: Solutions of second-order elliptic PDE problems on nonconvex or cracked domains with mixed Dirichlet-Neumann conditions typically lie outside...
By Somnath Giri, Muslim Malik, Rishikesh Yadav
arXiv:2603. 29237v2 Announce Type: replace Abstract: Enforcing prescribed global integral constraints in mesh-free neural PDE solvers is challenging in high-dimensional domains.
By Zhangyong Liang, Huanhuan Gao
arXiv:2606. 27895v1 Announce Type: cross Abstract: Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented.
By Andrin Rehmann, Heiko Zimmermann, Dion H\"afner
arXiv:2602. 02788v2 Announce Type: replace-cross Abstract: We aim to develop physics foundation models for science and engineering that provide real-time solutions to Partial Differential Equations (PDEs) which preserve structure and accuracy under adaptation to unseen geometries.
By Benjamin D. Shaffer, Shawn Koohy, Brooks Kinch, M. Ani Hsieh, Nathaniel Trask
arXiv:2609.08034v1 Announce Type: new
Abstract: Localized dimensionality reduction improves the scalability of operator learning for high-dimensional partial differential equations (PDEs), but indepe...
By Mrigank Dhingra, Jordan Stout, Omer San
arXiv:2606. 31342v1 Announce Type: cross Abstract: Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error.
By Haixin Wang, Haoning Dang, Fei Wang, Shimin Guo