arXiv Machine Learning
6d ago

Meshless Latent Dynamics Models for Model Order Reduction of time-dependent Partial Differential Equations

The paper introduces meshless Latent Dynamics Models (LDMs) for model order reduction of time‑dependent PDEs, using conditioning in Neural Fields to represent spatially continuous dynamics. It presents a unified mathematical framework, proves a Universal Approximation Theorem, and analyzes stability through Lipschitz bounds for different conditioning mechanisms, revealing an expressivity–stability trade‑off. Experiments on Burgers’ equation, Navier–Stokes flow, and linear advection demonstrate that meshless LDMs achieve comparable or better predictive accuracy than mesh‑based models while using only 2% of the parameters and maintaining near‑constant error across spatial discretizations.

By Lorenzo Tomada, Federico Pichi, Gianluigi Rozza
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.

By Lucas Gerken Starepravo, Henry Broadley, Steven Lind, Jack R. C. King