# Credits and contributions

## Original research and pretrained software

Blaine Quackenbush and Paul J. Atzberger developed the Geometric Neural Operators research underlying the pretrained-model demonstrations. Credit for the GNP methodology, upstream implementation, and pretrained models belongs to them and the contributors identified in their repository.

- Quackenbush, Blaine, and Paul J. Atzberger. Geometric Neural Operators (2024). https://doi.org/10.1088/2632-2153/ad8980 — open manuscript: https://arxiv.org/abs/2404.10843
- Quackenbush, Blaine, and Paul J. Atzberger. Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators (2025). https://doi.org/10.1088/2632-2153/ae1bf8 — open manuscript: https://arxiv.org/abs/2503.04649
- Quackenbush, Blaine, and Paul J. Atzberger. Extending Neural Operators: Robust Handling of Functions Beyond the Training Set (2026 preprint). https://arxiv.org/abs/2603.03621
- geo_neural_op source and pretrained weights: https://github.com/atzberg/geo_neural_op — GNU GPL version 3. Revision used: 4ba306ac20f5674b773a8ba980aabeef33ed5385. The software and weights are installed separately, not redistributed in this website package.

The six recorded neural prediction sets were computed using these upstream pretrained models. Their provenance, model names, weight hashes, inputs, and evaluation conditions accompany the data. Attribution of the website does not transfer authorship of the neural models or their predictions to the website developer.

## Website adaptation and completed enhancements — Joshua Randall

Website adaptation, implementation, and enhancements by Joshua Randall:

- Interactive 3D point-cloud viewer with pointer and keyboard controls.
- Responsive navigation, light/dark themes, and accessible text explanations.
- Live local polynomial curvature estimator and analytic reference comparisons.
- Six-case evaluation presentation with error measurements and reproducibility records.
- Interactive exact sphere-diffusion demonstration.
- Bayesian sphere-radius inference demonstration and posterior visualization.
- Point-cloud import/export and local experiment notebook.
- Companion Python workflow for generating and importing new GNP predictions.
- Guided experiments, scientific scope notes, integration instructions, and validation checks.

These are completed application and educational enhancements. The underlying curvature identities, PCA, weighted least squares, heat equation, and Bayesian inference are established mathematics. This project does not claim their invention, a new GNP architecture, or a demonstrated improvement to the authors' pretrained models. Comparative results are reported as measured, including the case where the conventional fit performed better.

## Licensing

Copyright © 2026 Joshua Randall for the original website implementation, companion scripts, and original explanatory material in this package, distributed under the included MIT license. External research, software, and pretrained weights retain their own authorship and license terms. The MIT license does not relicense the authors' papers or GPL-3.0 software and weights.