Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators

TL;DR

The paper introduces Geometric Neural Operators (GNPs) to learn operators on manifolds and other non-Euclidean domains by incorporating geometric descriptions directly into the learning process. The authors develop a training framework with lifting, trainable integral kernels, and projection layers, and they demonstrate graph-based approximations for integral operators and kernel factorization to improve efficiency. They apply GNPs to (i) estimate geometric quantities from point clouds, (ii) learn solution maps for Laplace-Beltrami PDEs on manifolds, and (iii) perform Bayesian inverse problems to identify manifold shapes from LB responses. Across experiments on radial manifolds and point-cloud data, GNPs achieve robust geometric quantity estimation, accurate LB solution operators, and effective shape inference, highlighting their potential for data-driven PDEs and geometry-aware modeling on complex geometries.

Abstract

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

Figures (6)

• Figure 1: Deep Learning Methods for Operators. An operator layer is used as part of processing input functions. For a function $v(\cdot)$, an affine operation is performed based on integration against a kernel $k(x,y)$ and adding a bias $b(\cdot)$. An additional skip connection with a local linear operator $W$ is also added to the pre-activation output of the layer. These linear operations are then followed by applying the activation function $\sigma(\cdot)$(left). In combination with the operations of lifting $\mathcal{P}$ and projection $\mathcal{Q}$, these layers are stacked to process input functions to obtain deep learning methods for approximating operators (right).
• Figure 2: Approximating $\mathcal{K}[v]$: Graph-based Approaches and Truncations. We develop methods for general manifolds, including with point-cloud representations (middle). We approximate integral operations by using graph neural operators and message passing. To evaluate $\mathcal{K}[v](x)$ at node $x$, we use mean aggregation. The nodes $x$ and $y$ have an edge connection in the graph when $\|x - y\| < r$. The graph has node attributes $v(y)$ and edge attributes $k(x,y)$. We take the mean over all updates $k(x,y)v(y)$ for neighboring nodes $\mathcal{N}(x)$ to obtain the approximation of $\mathcal{K}[v]$. To help make computations more efficient, we also truncate the neighborhood to a ball $B_r(x)$ of radius $r$(left). In deep learning methods, stacking the operator layers increases successively in depth from the previous layers the effective domain of dependence of the overall operator (right).
• Figure 3: Geometric Quantities for Manifolds with Point-Cloud Representations. We develop geometric neural operators (GNPs) to estimate geometric quantities, such as the metric and curvatures, from manifolds with point-cloud representations. Shown are a few example shapes and their Gaussian curvatures (left). The methods are trained on random shapes obtained by using barycentric coordinates to combine a collection of reference manifolds $A,B,C$ depicted at the vertices with functional forms given in Atzberger2018b(right).
• Figure 4: Training Geometric Operators for Solving PDEs on Manifolds. We train neural geometric operators (GNPs) for the solution map of the Laplace-Beltrami PDE. For the reference manifolds $A,B,C$ we show some example right-hand-sides (rhs) $f(x)$ and the corresponding solutions $u(x)$. We train with locations of the Gaussian for the rhs varied over the surface and over the collection of shapes.
• Figure 5: Manifold Shape Estimation: Prior Distribution. The prior distribution over manifolds $\mathcal{M}$ based on the radial shape covariances when $\beta = 1$. This is used in combination with Bayes' Rule and the geometric neural operators (GNPs) to obtain a posterior distribution over manifold shapes.