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

• URL: http://arxiv.org/abs/2404.10843v1
• Date: Tue, 16 Apr 2024 18:43:27 GMT
• Title: Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators
• Authors: Blaine Quackenbush, Paul J. Atzberger,
• Abstract summary: 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 to estimate geometric properties, such as the metric and curvatures, and to approximate Partial Differential Equations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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