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Documentation | Examples | Paper 1 | Paper 2 | Paper 3

Geometric Neural Operators (GNPs)

GNPs enable deep learning of features from point clouds and geometric datasets. Provides data-driven tools for learning and evaluating differential operators and solving PDEs on manifolds.

🟣 Core Functionality

  • Geometric Feature Learning: Extract intrinsic properties directly from point-cloud representations.
  • Differential Operators: Evaluate local curvatures and Laplace-Beltrami operators using learned protocols.
  • Advanced PDE Solvers: Robust solvers for Partial Differential Equations (PDEs) on manifolds.
  • Transferable Pretrained Models: Available models for immediate use with pretrained weights, see examples section.

🔵 Additional Functionality

  • Shape Evolution: Support for mean-curvature shape flows and other dynamic geometric tasks.
  • Efficient Architectures: Implementations using sparsity and factorizations for efficiency and trade-offs.
  • Modular Architecture: Easily integrate GNP components into larger data-processing pipelines.

Robust Estimation Approach

Our pre-trained GNP models and training methods also allow for coping with noise and other artifacts that arise when processing point-clouds. 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.

examples folder.

📦 Installation

git clone git@github.com:atzberg/geo_neural_op.git
conda create -n gnp python=3.12
conda activate gnp

Install PyTorch prior to installing the repo to avoid installation errors related to torch-cluster and torch-scatter. To install PyTorch with cpu, use:

pip install torch==2.6.0 --index-url https://download.pytorch.org/whl/cpu

To install with CUDA, use one of the following, replacing X with the correct version:

# CUDA 11.X
pip install torch==2.6.0 --index-url https://download.pytorch.org/whl/cu11X
# CUDA 12.X
pip install torch==2.6.0 --index-url https://download.pytorch.org/whl/cu12X

For available CUDA versions, see PyTorch Previous Versions. Once PyTorch is installed, you can install this repository using pip:

cd geo_neural_op
pip install .

If you want to run the example notebooks, you can install the additional dependencies using:

pip install .[dev]

If there is an error installing because of torch-cluster or torch-scatter, adding the flag --no-build-isolation should fix this. Building wheels for torch-cluster and torch-scatter can be quite time consuming.

Alternatively, you can install torch-cluster and torch-scatter separately using the pre-built wheels corresponding to your PyTorch installation. After they are installed, you can proceed with installing this repository. Installing using the pre-built wheels can be done using the appropriate command below:

# CPU Build
pip install torch_scatter torch_cluster -f https://data.pyg.org/whl/torch-2.6.0+cpu.html
# CUDA 11.X
pip install torch_scatter torch_cluster -f https://data.pyg.org/whl/torch-2.6.0+cu121X.html
# CUDA 12.X 
pip install torch_scatter torch_cluster -f https://data.pyg.org/whl/torch-2.6.0+cu12X.html

🤖 Testing

You can run tests for the package using

python -m unittest discover tests

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.

💡 Usage

For information on how to use the package, see

Version 2.0 Efficiency Gains

geo_neural_op v2.0.0 sees significant efficiency gains by leveraging the optimized data processing of PatchTensor and inference of the PatchGNP. This uses the new separable, block-factorized kernels (see papers). Below, we display a comparison of the average running times for versions 1.0.0 and 2.0.0 on both CPU and CUDA devices. Each task is run 10 times on each of the example data sets found in geo_neural_op/example_data. We display the average times to perform each task on one data sample from example_data. In all cases, v2.0 sees about >18x speed up on CPU and about a >7x speed up on CUDA devices.

Task v1.0.0 CPU v1.0.0 CUDA v2.0.0 CPU v2.0.0 CUDA
Geometric Quantities 139.39s 22.27s 7.55s 2.86s
Stiffness Matrix Construction 2,591.73s 524.32s 117.85s 7.85s
Mean Flow (10 steps) 1,484.20s 242.42s 81.61s 28.10s

📚 Additional Information

For the package, please cite:

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}
}

Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators, B. Quackenbush and P. J. Atzberger, Machine Learning: Science and Technology, 6.4, 045045, (2025), paper, 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, Paul},  
  journal={Machine Learning: Science and Technology},
  month = {11},  
  volume = {6},
  number = {4},
  pages = {045045},
  url={https://doi.org/10.1088/2632-2153/ae1bf8}
  publisher = {IOP Publishing},
  year={2025},
}

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

Additional Information
https://web.atzberger.org

Documentation | Examples | Paper 1 | Paper 2 | Paper 3