MEEC-Net: Meshfree Exterior Calculus for Data-Efficient Physics Learning
Researchers have introduced Meshfree Exterior Calculus (MEEC), a novel framework for learning structure-preserving physics descriptions directly from point clouds, eliminating the need for traditional mesh generation. This method underpins MEEC-Net, a data-efficient surrogate model capable of transferring across varying resolutions, geometries, and physical parameters. By equipping an epsilon-ball graph with virtual node and edge measures via a sparse Schur complement solve, MEEC ensures exact discrete conservation and end-to-end differentiability. The associated MEEC-Net learns unknown physics as a shared edge-wise flux law within an SO(d)-invariant local frame. Theoretical proofs establish a solution-error bound independent of problem geometry, explaining its strong generalization from limited training data. Empirical results on five canonical PDE benchmarks demonstrate that MEEC-Net achieves one to two orders of magnitude lower out-of-distribution error compared to baseline neural-operator approaches. Additionally, it shows competitive performance on the SimJEB structural-bracket benchmark while requiring substantially fewer training geometries, marking a significant advancement in generalizable and data-efficient scientific machine learning.
Wire timeline
MEEC-Net: Meshfree Exterior Calculus for Data-Efficient Physics Learning
Researchers have introduced Meshfree Exterior Calculus (MEEC), a novel framework for learning structure-preserving physics descriptions directly from point clouds, eliminating the need for traditional mesh generation. This method underpins MEEC-Net, a data-efficient surrogate model capable of transferring across varying resolutions, geometries, and physical parameters. By equipping an epsilon-ball graph with virtual node and edge measures via a sparse Schur complement solve, MEEC ensures exact discrete conservation and end-to-end differentiability. The associated MEEC-Net learns unknown physics as a shared edge-wise flux law within an SO(d)-invariant local frame. Theoretical proofs establish a solution-error bound independent of problem geometry, explaining its strong generalization from limited training data. Empirical results on five canonical PDE benchmarks demonstrate that MEEC-Net achieves one to two orders of magnitude lower out-of-distribution error compared to baseline neural-operator approaches. Additionally, it shows competitive performance on the SimJEB structural-bracket benchmark while requiring substantially fewer training geometries, marking a significant advancement in generalizable and data-efficient scientific machine learning.
cs.AI updates on arXiv.org