arXiv:2605. 13834v2 Announce Type: replace-cross Abstract: In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective.
arXiv:2606. 09806v1 Announce Type: cross Abstract: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains.
arXiv:2602. 02788v2 Announce Type: replace-cross Abstract: We aim to develop physics foundation models for science and engineering that provide real-time solutions to Partial Differential Equations (PDEs) which preserve structure and accuracy under adaptation to unseen geometries.
arXiv:2608.29892v1 Announce Type: new Abstract: Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scient...
arXiv:2609.05860v1 Announce Type: new Abstract: Neural operators that accept arbitrary meshes are often treated as geometry-general, but unseen domain topology changes both the invariant and decaying...
Combinatorial Network-Based Manifold Topological Deep Learning (CNMTDL) is a new framework that represents medical images as discrete manifolds and decomposes them into three Hodge components. Features from these components are concatenated and fed into a combinatorial complex architecture, enabling higher‑order message passing between 0‑cells and 2‑cells via attention‑based blocks. CNMTDL was evaluated on six 2D and 3D datasets from the MedMNIST v2 benchmark, showing improved performance for medical image analysis.
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.
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:2608. 14556v1 Announce Type: new Abstract: Physical fields on meshes require a separation between topology and geometry: conservation laws are topological and should be exact, while geometry, material response, and anisotropic coupling must be learned from data.
arXiv:2606. 16990v1 Announce Type: new Abstract: While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales.
arXiv:2607. 23192v1 Announce Type: cross Abstract: We develop a data-driven approximation of the projected ambient connection Laplacian acting on differential forms over smooth Riemannian manifolds sampled by point clouds.
The paper investigates how the Brinkman penalisation method, which embeds complex domain boundary-value problems into a simple computational box, preserves a multi-conformal symplectic structure for multi-symplectic Hamiltonian PDEs under a specific compatibility condition. It demonstrates that this leads to an exact local conservation law where the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. Building on these findings, the authors propose structure-preserving numerical integrators via Strang splitting and conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, and validate their approach with numerical experiments on wave and electromagnetic scattering.
arXiv:2509. 24627v2 Announce Type: replace Abstract: Embedding physical intuition into network architectures allows the learning of dynamics that enforce fundamental properties, such as energy conservation laws, thereby leading to physically-plausible predictions.