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

2026-05-08

AI Summary

This work addresses the challenge of structure-preserving, generalizable, and data-efficient learning of physical laws on mesh-free point clouds by introducing the Mesh-free Exterior Calculus (MEEC) framework. MEEC is the first to integrate structure-preserving discrete exterior calculus into point cloud learning, constructing differentiable complexes via ε-neighborhood graphs that satisfy discrete conservation laws. Built upon SO(d)-invariant local coordinate frames, the proposed MEEC-Net learns physical dynamics in the form of edge fluxes. The framework offers end-to-end differentiability, a rigorous error decomposition theory, and supports transfer across resolutions, geometries, and parameters. Evaluated on five canonical PDE benchmarks, MEEC achieves out-of-distribution errors one to two orders of magnitude lower than baseline methods, and demonstrates competitive accuracy on the SimJEB structural scaffold task with only limited training samples.

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.

Problem

• meshfree
• exterior calculus
• point clouds
• physics learning
• generalization

Innovation

• meshfree exterior calculus
• structure-preserving learning
• point cloud physics
• data-efficient surrogate
• SO(d)-invariant flux