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

TL;DR

This paper introduces a meshfree exterior calculus framework for physics learning from point clouds, enabling structure-preserving, transferable, and data-efficient surrogate models with strong out-of-distribution performance.

Contribution

It develops MEEC, a meshfree discretization that preserves physics structure, and MEEC-Net, a neural network leveraging this discretization for transferable physics learning.

Findings

MEEC-Net achieves 1-2 orders of magnitude lower out-of-distribution error than neural-operator baselines.

Single-solution training transfers effectively to unseen geometries and parameters.

On five PDE benchmarks, MEEC-Net outperforms traditional neural operators in accuracy and data efficiency.

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.