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Geometric Neural Operators (GNPs)

Geometric Neural Operators (GNPs) allow for data-driven deep learning of features from point-cloud representations and other datasets for tasks involving geometry. This includes training protocols and learned operators for estimating local curvatures, evaluating geometric differential operators, solvers for PDEs on manifolds, mean-curvature shape flows, and other tasks. The package provides practical neural network architectures and factorizations for training to accounting for geometric contributions and features. The package also has a modular design allowing for use of GNPs within other data-processing pipelines. Pretrained models are also provided for estimating curvatures, Laplace-Beltrami operators, components for PDE solvers, and other geometric tasks.

Robust Estimators: Our pre-trained GNP models and training methods also allow for coping with noise and other artifacts that arise when processing point-clouds in practice. This allows for robust estimates of the curvature and other geometric properties even when point-clouds have artifacts, such as outliers as shown below.

Examples: We provide practical demonstrations for how GNPs can be used in practice. This includes examples (i) to estimate geometric properties, such as the metric and curvatures of surfaces, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to perform curvature-driven flows of shapes. These results show a few ways GNPs can be used for incorporating the roles of geometry into machine learning processing pipelines and solvers.

Quick Start

bash git clone git@github.com:atzberg/geo_neural_op.git conda create -n gnp conda activate gnp pip install -r requirements.txt You may also need to first install pip, bash conda install pip

For use of the package see the examples folder.
More information on the structure of the package also can be found on the documentation pages.

Packages

The pip install should automatically handle most of the dependencies. If there are issues, please be sure to install pytorch package version >= 2.0.0. The full set of dependencies can be found in the requirements.txt. You may want to first install pytorch package manually to configure it for your specific GPU system and platform.

Usage

For information on how to use the package, see

• Examples

• Documentation

Additional Information

For the package, please cite:

Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators, B. Quackenbush and P. J. Atzberger, arXiv, (2025), arXiv. @article{quackenbush_atzberger_gnp_transfer_2025, title={Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators}, author={Quackenbush, Blaine and Atzberger, PJ}, journal={arXiv:2503.04649}, url={https://arxiv.org/abs/2503.04649}, year={2025} }

Geometric Neural Operators (GNPs) for Data-Driven Deep Learning in Non-Euclidean Settings, B. Quackenbush and P. J. Atzberger, Machine Learning: Science and Technology, 5.4, 045033, (2024), paper, arXiv. @article{quackenbush_atzberger_gnps_2024, title={Geometric neural operators (gnps) for data-driven deep learning in non-euclidean settings}, author={Quackenbush, Blaine and Atzberger, PJ}, journal={Machine Learning: Science and Technology}, volume={5}, number={4}, pages={045033}, url={https://doi.org/10.1088/2632-2153/ad8980}, publisher={IOP Publishing}, year={2024} }

Acknowledgements This work was supported by NSF Grant DMS-1616353 and NSF-DMS-2306345.

Documentation | Examples | Paper 1 | Paper 2