MEEC-Net

Meshfree Exterior Calculus (MEEC) and MEEC-Net for structure-preserving learning on point clouds.

Code accompanying our paper A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds.

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 ε-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.

Key results

We show single shot effective physics recovery in the local flux model, which enables extrapolation over boundary conditions, velocities, and geometries from a single training sample.

We also demonstrate massively improved data efficiency compared to baseline direc-prediction surrogates.

What's in this repo

A self-contained implementation of the MEEC-Net forward model:

• MEEC discretization: equips an ε-ball graph with virtual node and edge measures via a sparse Schur complement solve, producing a discrete exterior calculus complex that satisfies conservation exactly and is differentiable through point positions.
• Learned flux kernel: an SO(d)-invariant edge-wise flux law (MeshlessNeW) that produces compatible fluxes on any point cloud whose features lie in the training range.
• Differentiable Newton solver: sparse Newton solver with implicit function theorem backward pass.

The current demo.py runs a simple Poisson example, which will be update to provide more general hooks to training and evaluation.

Repository layout

• src/meshless_dec.py: MEEC operator assembly on 2D/3D point clouds (node/edge measures, boundary geometry, Laplacian).
• src/model.py: MeshlessNeW — encoder, Lipschitz-constrained flux kernel, and learned source model.
• src/solver.py: differentiable sparse Newton solver with IFT backward pass.
• src/utils.py: graph construction, boundary geometry, FEEC masking/projection utilities.
• demo.py: minimal entrypoint — Poisson smoke test and optional short training loop (--learn).

Dependencies

Tested with:

• Python 3.10+
• PyTorch 2.0+
• NumPy
• SciPy

Optional (for DC-PSE with non-negative edge measures):

• osqp

Minimal install (CPU):

pip install torch numpy scipy

For GPU / CUDA PyTorch, install PyTorch from the official selector for your platform.

Quickstart

Run the included demonstration:

python demo.py

For a short training loop:

python demo.py --learn