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.
arXiv:2512. 19643v2 Announce Type: replace Abstract: Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.
arXiv:2607. 11974v1 Announce Type: cross Abstract: Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen.
arXiv:2609.15620v1 Announce Type: new Abstract: Adaptive meshes enable neural operators for partial differential equations (PDEs) to allocate spatial samples and computation according to local physic...
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:2509.06154v3 Announce Type: replace Abstract: Developing accurate, data-efficient surrogate models is central to advancing AI for Science. Neural operators (NOs), which approximate mappings bet...
arXiv:2610.08475v1 Announce Type: new Abstract: In many-query scenarios, data-driven surrogate models provide an efficient alternative to high-fidelity solvers for simulating physical systems governe...
arXiv:2410. 10137v5 Announce Type: replace Abstract: We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds.
arXiv:2606. 11650v1 Announce Type: new Abstract: Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation.
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...
The paper introduces Sensitivity‑Constrained Neural Operators (SC‑NOs), which augment standard neural operator training with sampled Jacobian supervision from differentiable solvers or discrete adjoints. By matching selected sensitivities during training, SC‑NOs improve forward prediction accuracy and significantly enhance gradient‑based inverse reconstruction for distributed fields. Experiments on advection–diffusion, RANS–Spalart–Allmaras, high‑dimensional gridded inputs, and a shallow‑water tsunami source‑inversion case demonstrate that SC‑NOs achieve a better accuracy–cost trade‑off and enable near‑real‑time wave‑propagation forecasting from sparse observations.
arXiv:2608. 08608v1 Announce Type: cross Abstract: Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate.
arXiv:2607. 07718v1 Announce Type: cross Abstract: Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations.