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: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.
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:2606. 19251v1 Announce Type: cross Abstract: Solving the pressure-Poisson equation remains the primary computational bottleneck in incompressible unstructured flow solvers primarily due to the inherent sensitivity of traditional linear solvers to mesh irregularities.
The paper presents a neural hierarchical‑matrix preconditioner designed for real‑time GPU solves of sparse symmetric positive‑definite systems that change each frame. By training a graph‑and‑attention network to predict an SPD approximate inverse in H²‑matrix format, the method achieves linear‑time inference and application, outperforming traditional multigrid setup times and local preconditioners. Experiments on 3D mesh diffusion problems show the preconditioner reduces conjugate‑gradient iterations from 116 to 33 and enables 120 fps real‑time performance for up to 3,647 unknowns.
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:2509. 08765v4 Announce Type: replace-cross Abstract: Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications.
arXiv:2607. 13566v1 Announce Type: cross Abstract: For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy.
arXiv:2509. 10378v2 Announce Type: replace-cross Abstract: Linear systems arise in generating samples and in calculating observables in lattice quantum chromodynamics~(QCD).
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:2608.21234v1 Announce Type: cross Abstract: These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classic...
arXiv:2607. 18020v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators.
arXiv:2609.36216v1 Announce Type: new Abstract: Neural operators are typically trained in a supervised fashion, which requires a dataset to be generated with a classical solver. Training them physics...