The paper introduces Riemannian Hodge Message Passing (RHMP), a neural network architecture that separates topology from geometry in physical field modeling on meshes. RHMP fixes the cellular coboundaries determined by mesh incidence and learns symmetric positive‑definite cochain metrics to guide geometry‑dependent propagation, ensuring exact cochain‑complex identities and cochain‑frame equivariance. Experiments on seven benchmarks—including fluids, electromagnetism, gauge fields, and variable‑mesh CFD—show that RHMP achieves strong performance, especially when topology, learned geometry, and field structure interact.
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. 13827v1 Announce Type: cross Abstract: Markovian Whittle-Mat\'ern fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}.
arXiv:2606. 13827v2 Announce Type: replace-cross Abstract: Markovian Whittle-Mat\'ern fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}.
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...
arXiv:2605. 00394v3 Announce Type: replace Abstract: We present Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net: a structure-preserving framework for mesh-based continuum physics that cleanly separates the physics' topological structure from its metric structure.
arXiv:2609.37817v1 Announce Type: new Abstract: Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori ($\mathbb{T}^K$). Howev...
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. 00937v1 Announce Type: new Abstract: Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data.
arXiv:2609.35948v1 Announce Type: cross Abstract: Geometry does more than constrain an associative memory: curvature determines what it remembers and which states it creates. We develop intrinsic den...
Methods operating on Vision Transformer (ViT) feature spaces typically rely on Euclidean distance or cosine similarity. This assumes that every direction is equally meaningful, but there is no reason...
arXiv:2607. 22215v1 Announce Type: new Abstract: In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data.
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.