arXiv:2606. 30495v1 Announce Type: cross Abstract: Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors.
arXiv:2608. 11572v1 Announce Type: new Abstract: Coarse-grid numerical solvers can substantially reduce the computational cost of time-dependent PDE simulation, but under-resolution often degrades both the trajectory and the spatial fidelity of the solution.
arXiv:2607. 28456v1 Announce Type: cross Abstract: Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning.
Coarse-grid numerical solvers can substantially reduce the computational cost of time-dependent PDE simulation, but under-resolution often degrades both the trajectory and the spatial fidelity of the solution. We introduce RECAST (Recurrent Error Correction And Super-resolution of coarse-grid Trajectories), a machine-learning framework designed to restore this lost accuracy while retaining coarse-grid evolution.
arXiv:2601. 20174v3 Announce Type: replace-cross Abstract: Solving large-scale sparse linear systems originating from partial differential equations (PDEs) is a fundamental topic in high-performance scientific computing, where preconditioners are crucial.
arXiv:2508. 20650v2 Announce Type: replace Abstract: Addressing the computational challenges of high-frequency and multiscale partial differential equations (PDEs), this work introduces a self-composing neural operator (SC-NO) framework.
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.
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.
arXiv:2604.06433v2 Announce Type: replace-cross Abstract: The impact of wave-induced forcing on the mean water level and nearshore currents is typically modeled through excess momentum fluxes, also k...
arXiv:2605. 26854v2 Announce Type: replace Abstract: The scalable solution of large sparse linear systems is a bottleneck in scientific computing and graph analysis.
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...
arXiv:2606. 18305v1 Announce Type: cross Abstract: Operator learning is an emerging interdisciplinary field that integrates machine learning with scientific computing.
arXiv:2606. 02582v1 Announce Type: cross Abstract: In this paper, we study the efficient numerical solution of Darcy equations in strongly heterogeneous media with high-contrast permeability and propose a hybrid framework that combines learning with multiscale numerical methods.