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

TL;DR

The paper addresses robust geometric analysis from point-cloud data and solving geometric PDEs on manifolds. It introduces transferable Geometric Neural Operators (GNPs) that map point-cloud samples to local parameterizations and latent geometric features using a lifting layer, kernel-integral layers, and a projection, trained on radial manifolds parameterized by spherical harmonics. Key contributions include noise-robust training, transfer to unseen topologies such as toroidal shapes, and applications to metric/curvature estimation, mean-curvature flow, and Laplace-Beltrami PDE solvers, with an open-source geo_neural_op package. This work provides data-driven, reusable geometric estimators and numerical solvers that can enhance pipelines in geometry-related inference and simulation tasks.

Abstract

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.

Figures (6)

• Figure 1: Geometric Neural Operators (GNPs). We learn features from point-clouds using geometric neural operators GNPs Quackenbush2024. These neural operators consist of three learnable components (i) a lifting operation $\mathcal{P}[u,\{\mathbf{x}_i\}]$ that provides initial features for the input geometry data $\{\mathbf{x}_i\}_{i=1}^N$ and input function $u(\cdot)$, (ii) layers of operators $\mathcal{O}^{(i)}$ that consist of kernel integral operations $\mathcal{K}[v](x)$ and affine operations $\left(\mathcal{W}v\right)(x) + b(x)$ each of which are passed through a non-linear activation operation $\sigma[\cdot]$, and (iii) a projection operation $\mathcal{Q}$ for constructing the final output function $w(x) = w(\xi^1,\xi^2)$ and local parameterization $\boldsymbol{\tilde{\sigma}}(\xi^1,\xi^2) = \mathbf{\bar{x}} + \xi^1 \psi_1 + \xi^2\psi_2$. We use geometric neural operators (GNPs) to map a collection of points $\{x_i\}_{i=1}^N$ sampled from the geometry to local parameterizations and functions $w = w(\xi^1,\xi^2)$.
• Figure 2: Learning Latent Geometric Representations. We show how data is processed by our geometric neural operators (GNPs). We obtain latent representations at each $\mathbf{\bar{x}}$ by learning GNPs that map a collection of points $\{\mathbf{x}_i\}_{i=1}^N$ sampling the geometry to a local parameterization $(\xi^1,\xi^2)$ and surface height function $s(\xi^1,\xi^2)$. This provides geometric information for further processing and tasks.
• Figure 3: GNP Estimators for Gaussian Curvatures on Test Shapes. Gaussian curvatures of radial manifold (left) and toroidal manifold (right). The views show the same manifold from a few different vantage points.
• Figure 4: Point-Cloud Data with Artifacts. The underlying geometry can be obscured by noise when working with point-cloud representations. We focus on the case of non-uniform samplings and outlier artifacts. For an example shape sampled with approximately $10\%$ outliers (left), we highlight a subset of the outliers in the data by circles (middle). The shape also exhibits non-uniform sampling as can be seen in the most magnified view (right). These and other artifacts can be introduced into our datasets for training to enhance the robustness of the GNP methods.
• Figure 5: Mean-Curvature Driven Flow. We show shape deformations evolved under mean-curvature flow (MCF) in equation \ref{['equ_mean_flow']} using our numerical methods based on pretrained GNPs.