MEEC-Net learns physics from single examples, transfers across geometries

Structure-preserving neural surrogates that generalize across geometries and resolutions could reshape how engineers build fast approximations for structural mechanics, fluid dynamics, and other PDE-governed systems—and a new approach achieves that transfer after training on as few as one example.

MEEC-Net, introduced in a preprint posted to arXiv on May 12, 2026, builds on meshfree exterior calculus (MEEC), a discretization framework that equips an ε-ball graph with virtual node and edge measures via a sparse Schur complement solve. The resulting complex satisfies discrete conservation exactly, is end-to-end differentiable in point positions, and bypasses the mesh-generation step required by conventional structure-preserving methods. The network 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.

The authors prove a solution-error bound that splits into discretization and kernel-approximation terms independent of problem geometry—a theoretical result that explains why the model transfers from very few examples. 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 matches competitive error while using substantially fewer training geometries. Single-solution training transfers to unseen geometries, boundary conditions, and physical parameters across all experiments.

The work demonstrates that structure-preserving discretizations can be learned end-to-end on point clouds without sacrificing the geometric guarantees that make them valuable for physics simulation. Full details are available in the arXiv preprint at arxiv.org/abs/2605.08436.