← Back to all news
arXiv Machine Learning

Learning Spectrally Optimised Mesh-Free Discretisations

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.

Related stories

arXiv Machine Learning
Sep 3

Learning Spectral-Like Mesh-Free Discretisations

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
More like this →
arXiv Machine Learning
Aug 28

Enforcing Dirichlet Boundary Conditions in Operator Learning

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
More like this →
arXiv Machine Learning
Jun 4

Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

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
benchmarks
More like this →
arXiv AI
2d ago

Neural networks for spectral optimization

arXiv:2609.36047v1 Announce Type: cross Abstract: Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional....

By Alexis de Villeroch\'e, Beniamin Bogosel, St\'ephane Breuils, Dorin Bucur, Jacques-Olivier Lachaud
More like this →