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

Researchers introduce MEEC (meshfree exterior calculus), a novel framework for learning physics directly from point clouds without requiring mesh generation. MEEC-Net, built on this approach, demonstrates 1-2 orders of magnitude better generalization across different geometries, resolutions, and physical parameters compared to existing neural operator methods, achieving this with minimal training data.

The introduction of meshfree exterior calculus represents a significant methodological advancement in scientific machine learning, addressing a fundamental challenge in neural operator design: generalizing learned physics models across different geometric domains. Traditional structure-preserving discretizations require explicit mesh construction, which introduces computational overhead and breaks generalization when geometry changes. MEEC bypasses this bottleneck by operating directly on point clouds through virtual node and edge measures computed via sparse Schur complement solvers, maintaining discrete conservation laws while remaining fully differentiable with respect to point positions.

This work emerges from the broader convergence of differential geometry and machine learning, where preserving mathematical structure during learning has proven crucial for both accuracy and generalization. Previous neural operator approaches (DeepONet, Fourier Neural Operators) achieved high accuracy within their training distribution but often failed dramatically on out-of-distribution problems. MEEC-Net's SO(d)-invariant local frame construction elegantly decouples learned physics from absolute coordinates, enabling a single trained kernel to work across arbitrary point cloud configurations where feature ranges align.

The practical implications extend across scientific computing, engineering simulation, and digital twin applications. The ability to train on single solutions and transfer to unseen geometries, boundary conditions, and parameters dramatically reduces data requirements—a critical advantage for expensive physical simulations. The competitive performance on the SimJEB structural-bracket benchmark while using substantially fewer training geometries demonstrates practical engineering utility. This approach could accelerate surrogate modeling for finite-element analysis, computational fluid dynamics, and topology optimization workflows where mesh-based methods currently dominate.

The theoretical proof bounding solution error independently of problem geometry provides mathematical grounding that justifies observed transfer capabilities, distinguishing this from purely empirical neural network approaches.

• →MEEC-Net achieves 1-2 orders of magnitude lower out-of-distribution error than existing neural operator baselines through geometry-agnostic learning
• →The framework transfers across resolutions, geometries, and physical parameters from minimal training data without explicit mesh generation
• →Discrete conservation laws are satisfied exactly through virtual node and edge measures via sparse Schur complement computation
• →SO(d)-invariant local frame design enables a single trained kernel to produce compatible physics on any point cloud configuration
• →Error bounds independent of problem geometry provide theoretical justification for observed generalization across diverse domains