Learning Spectrally Optimised Mesh-Free Discretisations

Numerical methods for partial differential equations (PDEs) discretise differential operators, ideally reproducing the action of the continuous operators across all wavenumbers permitted by a given discretisation. Spectral-type methods approach this ideal, but rely on structured grids or high-order meshes that are hard to generate for complex geometries. In contrast, mesh-free methods are geometrically flexible, yet how faithfully they reproduce the operator across resolved scales is strongly influenced by a heuristically chosen kernel, which selects one of many weight sets satisfying the same consistency conditions without regard to the resulting spectral response. To address this, we introduce Spectrally optimised Neural Discretisations (SpeND), a framework that learns the map from local stencil geometry to discretisation weights on unstructured point clouds. A projection layer constrains every predicted stencil to the affine set defined by the discrete moment conditions, so polynomial consistency, and hence formal order of accuracy, holds exactly. Since consistency is enforced by the architecture, the weights can additionally be optimised for accuracy on a prescribed function space. Herein, we use Fourier modes and target the exact differentiation response over a chosen wavenumber band, so training is unsupervised. The loss function is a design interface: changing how it weights wavenumbers yields operators with distinct accuracy profiles. The trained discrete differential operators are PDE-agnostic and are applied without retraining to the Poisson, Burgers and Navier--Stokes equations, to near-boundary stencils, and across resolutions. At the same order and resolution, SpeND matches or improves on established mesh-free discretisations, and has been shown to reduce the wall-clock time between 3-20× to achieve equivalent error as the baselines.