Partial Differential Equations

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May 8, 2026 · cs.LGJ/K move · Enter open · S save

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

Benjamin D. Shaffer, Brooks Kinch, M. Ani Hsieh, Nathaniel Trask

University of Pennsylvania Philadelphia, PA 19104 · *Mechanical Engineering and Applied Mechanics

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.

Meshes Partial Differential Equation Benchmarks Classical Numerical Solver

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