Euclidean Fourier Neural Operators (EFNOs) extend Fourier neural operators by making the spectral kernel a continuous function of physical wavevectors, thereby removing dependence on specific periodic domain shapes and sizes. This domain‑independent formulation allows EFNOs to learn operators that generalize across different grid resolutions and domain geometries. Experiments on a heat equation and a materials‑science task demonstrate that EFNOs can successfully transfer learned mappings to unseen grid sizes and crystal structures.
By Nathanael Bosch, Niklas Frederik Schmitz, Michael F. Herbst
arXiv:2606. 11518v1 Announce Type: cross Abstract: Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations.
By Pengqing Shi, Jie Yin, Stephen Tierney, Junbin Gao
arXiv:2608. 14733v1 Announce Type: cross Abstract: Building on the foundation of single-hidden-layer neural networks, Fourier Feature Networks (FENs) are proposed, which incorporate Fourier features using $\cos$, $\sin$, or a combination of both.
By Qihong Yang, Zhijie Su, Yangtao Deng, Qiaolin He
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:2606. 28122v1 Announce Type: cross Abstract: Neural operators provide deep neural networks for learning mappings between function spaces.
By Alex Colagrande, Paul Caillon, Eva Feillet, Alexandre Allauzen
arXiv:2606. 27459v1 Announce Type: new Abstract: We consider the cubic nonlinear Schr\"odinger (NLS) equation on two-dimensional flat tori with varying aspect ratios.
By Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu
arXiv:2606. 27459v2 Announce Type: replace Abstract: We consider the cubic nonlinear Schr\"odinger (NLS) equation on two-dimensional flat tori with varying aspect ratios.
By Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu
arXiv:2608. 08608v1 Announce Type: cross Abstract: Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate.
By Roberto Nuca, Giovanni Testa, Luca Galimberti, Matteo Parsani
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 a neural operator architecture that inherently satisfies homogeneous Dirichlet boundary conditions by constraining each layer’s output to lie within the span of selected Dirichlet eigenfunctions of the Laplacian. This design works for any bounded domain with a Lipschitz boundary and any discretization, avoiding the restrictions of previous methods. The authors prove universal approximation for their architecture and demonstrate its effectiveness on Darcy flow and Helmholtz equation problems.
By Andrew M. Stuart, Margaret Trautner
arXiv:2407. 00809v4 Announce Type: replace Abstract: This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from functions to functions).
By Matthew Lowery, John Turnage, Zachary Morrow, John D. Jakeman, Akil Narayan, Shandian Zhe, Varun Shankar
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