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

NeurIPS 2026 · Point clouds › Point cloud classification · approach: Equivariant point cloud networks · 0 citations (OpenAlex)

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Introduces MEEC, a meshfree discrete exterior calculus on epsilon-ball point-cloud graphs built via one sparse Schur solve, and MEEC-Net, which learns a shared SO(d)-invariant edge-wise flux law trained through the PDE solve for data-efficient, transferable surrogates.

Problem

Learning physics surrogates on point clouds that generalize across geometry, resolution, boundary conditions, and physical parameters from few training examples, without mesh generation and while preserving conservation structure.

Approach

A meshfree exterior calculus whose codifferential is obtained from a QP enforcing local polynomial reproduction via a sparse Schur complement solve, combined with a learned local edge-frame flux law inserted into a conservative discrete PDE solve so the network learns constitutive relations rather than global solution maps.

Key results

• On canonical PDE benchmarks with N_train=100, MEEC-Net reports OOD relative L2 errors of 0.57% (Darcy) to 2.68% (elasticity) versus best baseline OOD errors around 12.8%-52%, i.e., 1-2 orders of magnitude lower.
• On advection-diffusion, MEEC-Net error is nearly constant in training-set size, consistent with single-shot physics recovery from one training solution.
• The method recovers O(h^2) solution-level convergence on Poisson and geometry-extrapolated problems, while the proven truncation rate is O(h).

Surprising findings

• Some Hodge-star edge weights are negative, yet enforcing positivity showed no performance difference, so the default is the equality-only QP.
• The learned flux recovered from a single training solution generalizes to unseen geometries and boundary conditions without retraining.

Limitations

• Targets only steady-state elliptic problems; time integration omitted.
• Learned constitutive law restricted to local first-order class K(u, grad u); nonlocal or higher-derivative closures not covered.
• Bounds assume quasi-uniform point clouds with bounded edge degree; anisotropic distributions not analyzed.

Experimental setup

Datasets: SimJEB

Benchmarks: advection-diffusion Darcy flow linear elasticity nonlinear advection-diffusion nonlinear Darcy SimJEB

Metrics: normalized L2 relative error convergence order

Baselines: PointNet GNN MeshGraphNet Transolver UPT Linear Attention Galerkin Attention DeepJEB-trained methods

Builds on: discrete exterior calculus Mirzaei local polynomial reproduction theory DEC-based data-driven learning on meshes

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