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We develop data-driven methods incorporating geometric and topological information to learn parsimonious representations of nonlinear dynamics from observations. The approaches learn nonlinear state-space models of the dynamics for general manifold latent spaces using training strategies related to Variational Autoencoders (VAEs). Our methods are referred to as Geometric Dynamic (GD) Variational Autoencoders (GD-VAEs). We learn encoders and decoders for the system states and evolution based on deep neural network architectures that include general Multilayer Perceptrons (MLPs), Convolutional Neural Networks (CNNs), and other architectures. Motivated by problems arising in parameterized PDEs and physics, we investigate the performance of our methods on tasks for learning reduced dimensional representations of the nonlinear Burgers Equations, Constrained Mechanical Systems, and spatial fields of Reaction-Diffusion Systems. GD-VAEs provide methods that can be used to obtain representations in manifold latent spaces for diverse learning tasks involving dynamics.
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This content will become publicly available on November 6, 2025
Geometric neural operators (gnps) for data-driven deep learning in non-euclidean settings
We introduce Geometric Neural Operators (GNPs) for data-driven deep learning of geometric features for tasks in non-euclidean settings. We present a formulation for accounting for geometric contributions along with practical neural network architectures and factorizations for training. We then demonstrate how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures of surfaces, (ii) to approximate solutions of geometric partial differential equations on manifolds, and (iii) to solve Bayesian inverse problems for identifying manifold shapes. These results show a few ways GNPs can be used for incorporating the roles of geometry in the data-driven learning of operators.
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- Award ID(s):
- 2306101
- PAR ID:
- 10611638
- Publisher / Repository:
- IOP Science
- Date Published:
- Journal Name:
- Machine Learning: Science and Technology
- Volume:
- 5
- Issue:
- 4
- ISSN:
- 2632-2153
- Page Range / eLocation ID:
- 045033
- Subject(s) / Keyword(s):
- Machine Learning Neural Operator Deep Learning Geometric Learning Differential Geometry Partial Differential Equations Bayesian Learning
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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