# Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators

arXiv: https://arxiv.org/abs/2503.04649 · HTML: https://mlsanity.com/paper/2503.04649/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 88 references that are in the map.
- First posted on arXiv 6 Mar 2025.

## Builds on (most cited first, 30 of 36)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Understanding Batch Normalization](https://mlsanity.com/paper/1806.02375/) (2018)
- [Semantic3D.net: A new Large-scale Point Cloud Classification Benchmark](https://mlsanity.com/paper/1704.03847/) (2017)
- [Deep Learning for LiDAR Point Clouds in Autonomous Driving: A Review](https://mlsanity.com/paper/2005.09830/) (2020)
- [BERT: A Review of Applications in Natural Language Processing and Understanding](https://mlsanity.com/paper/2103.11943/) (2021)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Designing Deep Networks for Surface Normal Estimation](https://mlsanity.com/paper/1411.4958/) (2014)
- [AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformers](https://mlsanity.com/paper/2502.09692/) (2025)
- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)
- [Diffeomorphism Neural Operator for various domains and parameters of partial differential equations](https://mlsanity.com/paper/2402.12475/) (2024)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Finsler-Laplace-Beltrami Operators with Application to Shape Analysis](https://mlsanity.com/paper/2404.03999/) (2024)
- [Point-DeepONet: Predicting Nonlinear Fields on Non-Parametric Geometries under Variable Load Conditions](https://mlsanity.com/paper/2412.18362/) (2024)
- [Learning the Geodesic Embedding with Graph Neural Networks](https://mlsanity.com/paper/2309.05613/) (2023)
- [Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach](https://mlsanity.com/paper/1905.10469/) (2019)
- [GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductions](https://mlsanity.com/paper/2206.05183/) (2022)

## Built on by (most cited first, 1 of 1)

- [Extending Neural Operators: Robust Handling of Functions Beyond the Training Set](https://mlsanity.com/paper/2603.03621/) (2026)

## Abstract (arXiv)

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.