Related papers: A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds

A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds

URL: http://arxiv.org/abs/2605.08436v1
Date: Fri, 08 May 2026 19:59:40 GMT
Title: A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds
Abstract summary: We introduce a meshfree exterior calculus (MEEC) for learning structure-preserving descriptions of physics on point clouds.<n>We use it to build MEEC-Net, a data-efficient surrogate that transfers across resolutions, geometries, and physical parameters.
Score: 8.15594797043476
License: http://creativecommons.org/licenses/by/4.0/
Abstract: We introduce a meshfree exterior calculus (MEEC) for learning structure-preserving descriptions of physics on point clouds, and use it to build MEEC-Net, a data-efficient surrogate that transfers across resolutions, geometries, and physical parameters. MEEC equips an $\varepsilon$-ball graph with virtual node and edge measures via a single sparse Schur complement solve; the resulting complex satisfies discrete conservation exactly, is end-to-end differentiable in the point positions, and exposes a direct geometry-to-physics link without the mesh-generation step required by conventional structure-preserving discretizations. MEEC-Net learns unknown physics as a shared edge-wise flux law in an SO($d$)-invariant local frame, so the same kernel produces compatible fluxes on any point cloud whose features lie in the training range. We prove a solution-error bound that splits into discretization and kernel-approximation terms which is independent of problem geometry, explaining the observed transfer from very few examples. We show that single-solution training transfers to unseen geometries, boundary conditions, and physical parameters. On five canonical PDE benchmarks MEEC-Net achieves 1-2 orders of magnitude lower out-of-distribution error than baseline neural-operator approaches. On the SimJEB structural-bracket benchmark it achieves competitive error while using substantially fewer training geometries.

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