arXiv Machine Learning
Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains
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SFO: Learning PDE Operators via Spectral Filtering
arXiv:2601. 17090v2 Announce Type: replace-cross Abstract: Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps.
arXiv Machine Learning
Aug 31
Beyond Pairwise Graphs in Science: Hypergraph Adaptive Wavelet Operators for Parametric PDEs
arXiv:2608.27883v1 Announce Type: new Abstract: Physical systems are often modeled by solution operators that map input fields, parameters, geometries, or past states to steady or future physical sta...
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EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs
arXiv:2606. 03260v1 Announce Type: cross Abstract: Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems.
arXiv Machine Learning
Sep 30
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.
arXiv AI
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From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning
arXiv:2608. 06894v1 Announce Type: new Abstract: Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations.
arXiv Machine Learning
Sep 30
GRIFDIR: Graph Resolution-Invariant Diffusion Models over Irregular Domains
GRIFDIR is a new architecture for score-based diffusion models that operates directly on unstructured meshes, enabling function-space diffusion over irregular domains. By representing generalized graph convolutional kernels as finite element functions, the model achieves resolution invariance and can handle complex, non-convex, and multiply-connected geometries. Experiments demonstrate that GRIFDIR maintains high fidelity in both unconditional and conditional sampling across diverse shapes.