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: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
arXiv:2606. 17460v1 Announce Type: new Abstract: Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows.
By Lennon J. Shikhman
arXiv:2507.07292v2 Announce Type: replace
Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
By Jacob Hauck, Yanzhi Zhang
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
arXiv:2609.35938v1 Announce Type: new
Abstract: This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitl...
By Yuan Guo, Hanshu Chen, Qiang Xi, Timon Rabczuk, Zhuojia Fu
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
By Heechang Kim, Qianying Cao, Hyomin Shin, Seungchul Lee, George Em Karniadakis, Minseok Choi
arXiv:2608. 06428v1 Announce Type: new Abstract: Deep Operator Networks (DeepONets; arXiv:1910.
By Khemraj Shukla, George Em Karniadakis
arXiv:2601. 20496v2 Announce Type: replace-cross Abstract: Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications.
By Ofek Aloni, Barak Fishbain
arXiv:2606. 11963v1 Announce Type: new Abstract: Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space.
By Mostafa Bamdad, Mohammad Sadegh Eshaghi, Timon Rabczuk
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
The paper introduces Spectral-like Neural Discretisation (SpeND), a mesh‑free method that learns stencil weights via a neural network to approximate the modal response of a spectral operator across a specified band of wavenumbers. By projecting the network output onto the space of polynomial‑consistent weights, SpeND ensures exact consistency while minimizing dispersion and dissipation errors in a self‑supervised, physics‑agnostic manner. Experiments on disordered 2‑D node sets demonstrate that the learned fourth‑order operator matches the exact spectral response over a wider band than traditional LABFM or structured‑grid finite differences, and retains fourth‑order convergence upon refinement.
By Lucas Gerken Starepravo, Henry Broadley, Steven Lind, Jack R. C. King