arXiv:2606. 18175v1 Announce Type: cross Abstract: We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization.
By Gbenga T. Awojinrin, Abdul-Akeem Olawoyin, Rami M. Younis
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. 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
arXiv:2510. 10350v3 Announce Type: replace-cross Abstract: Operator learning provides a data-driven approach to approximating solution operators of partial differential equations, but its effectiveness depends strongly on how input and output functions are represented.
By Chuqi Chen, Yang Xiang, Weihong Zhang
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
Transolver‑σ is a neural PDE solver that jointly models spectral and physical subspaces to improve accuracy in both one‑step and autoregressive rollouts. The method uses adaptive physical-state interactions, Slice‑Residual Physics‑Attention, and an axis‑factorized Fourier operator to enable information exchange between representations. Across five standard PDE benchmarks, Transolver‑σ reduces benchmark‑averaged relative error by 33.4% compared to the strongest baseline and shows strong performance on coupled multiphysics systems and real‑world fluid and combustion data.
By Haonan Shangguan, Hang Zhou, Haixu Wu, Yuezhou Ma, Jianmin Wang, Mingsheng Long
arXiv:2609.24947v1 Announce Type: new
Abstract: Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed n...
By S. Mohammad Mousavi, Teeratorn Kadeethum, Nikolaos Bouklas, Somdatta Goswami
arXiv:2607. 25608v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks.
By Pinki Khatun, M. Sajid, Abhinav Jha, M. Tanveer
arXiv:2606. 12337v1 Announce Type: cross Abstract: Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative.
By Zhen Zhang, Alessandro Alla, George Em Karniadakis
arXiv:2603. 13751v2 Announce Type: replace Abstract: Physics-informed neural networks (PINNs) have achieved notable success in modeling dynamical systems governed by partial differential equations (PDEs).
By Zhangyong Liang, Huanhuan Gao
arXiv:2606. 21828v2 Announce Type: replace-cross Abstract: Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction.
By Jaemin Oh, Youngkyu Lee, Jerome Darbon, George Em Karniadakis
arXiv:2609.22189v1 Announce Type: cross
Abstract: The Schrodinger equation in one spatial dimension admits a small set of exactly solvable potentials that serve as natural proving grounds for any new...
By Tariq Mahmood, Waqas Arshad, Bilal Naseer, Alfredo Raya