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:2510. 19399v2 Announce Type: replace Abstract: Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs).
By Yulun Wu, Miguel Aguiar, Karl H. Johansson, Matthieu Barreau
arXiv:2602.08515v3 Announce Type: replace-cross
Abstract: This work investigates shallow physics-informed neural networks (PINNs) for solving forward and inverse problems governed by nonlinear partia...
By Muhammad Luthfi Shahab, Imam Mukhlash, Hadi Susanto
The paper introduces a linearized Physics-Informed Neural Network (lPINN), a reduced‑order neural basis approach for solving forward and inverse differential equations. In an offline phase, lPINN learns continuous, differentiable neural basis functions from numerical solutions, which are then frozen for new problem instances; the online solution is obtained by minimizing the governing‑equation residual with additional constraints. Experiments on advection‑diffusion, Burgers', and nonlinear pendulum equations show that lPINN achieves lower solution and parameter errors than vanilla PINNs while reducing online inference times by up to three orders of magnitude, and its continuous representation generalizes to finer meshes without retraining.
By Wenhao Chen, Alexandre M. Tartakovsky
arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.
By Zhiwen Yu, Derong Yang, Liujian Zhang, Kaixiang Yang, Peilin Zhan, Jianmin Lv, Jane You, C. L. Philip Chen
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. 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
The paper proposes Domain-aware Fourier Features (DaFFs) for Physics-Informed Neural Networks (PINNs), embedding domain-specific geometry and boundary conditions into the positional encoding. DaFFs eliminate the need for explicit boundary loss terms, simplify optimization, and reduce training cost, leading to orders-of-magnitude lower errors and faster convergence compared to vanilla PINNs and RFF-based PINNs. An LRP-based explainability framework further shows that DaFFs produce more physically consistent relevance attributions, improving interpretability.
By Alberto Mi\~no Calero, Luis Salamanca, Konstantinos E. Tatsis
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).
By Qihong Yang, Qiaolin He
The paper introduces the Physics-Informed Stochastic Configuration Machine (PI‑SCM), a backpropagation‑free neural network designed for solving nonlinear differential equations. By analytically evaluating local Jacobians, PI‑SCM linearizes the physical loss, enabling optimal weight determination through generalized linear least squares and avoiding iterative nonlinear optimization. The authors present a progressive algorithmic suite—PI‑SC‑I, PI‑SC‑II, and PI‑SC‑III—prove their universal approximation properties, and show through experiments that PI‑SCM achieves high‑fidelity predictions and parameter identification while accelerating training by orders of magnitude compared to standard PINNs.
By Yuehao Song (School of Automation, Central South University, Changsha, China), Zhong Chen (School of Automation, Central South University, Changsha, China), Lihui Cen (School of Automation, Central South University, Changsha, China), Liang Wu (Johns Hopkins University, Baltimore, USA), Kai Zhang (State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing, China)
arXiv:2607. 02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers.
By Joseph Webb, Sadok Jerad, Coralia Cartis
arXiv:2607. 23940v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution.
By Pavlos Protopapas, Kaylee Vo