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arXiv · 2606.23834

Efficient model-order reduction via graph autoencoders and operator learning for systems with sharp gradients using a novel point cloud error metric for performance assessment

Liam K Magargal·Parisa Khodabakhshi
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Abstract

This study investigates the use of graph autoencoders, a class of graph neural networks (GNNs), to perform model-order reduction (MOR) on high-dimensional systems characterized by sharp gradients for which conventional linear approximations, such as proper orthogonal decomposition (POD) bases, are less effective. Specifically, this study introduces graph neural network latent space dynamics identification (GNN-LaSDI), a MOR framework that uses an operator learning strategy to directly evaluate the temporal evolution of the graph autoencoder's latent representation. In addition to using standard error metrics, this study presents a novel point cloud error metric tailored to directly evaluate the locations of sharp gradients, such as shocks in compressible fluid simulations and solid-liquid interfaces in phase-field simulations. The performance of GNN-LaSDI is compared to that of geometric deep least-squares Petrov-Galerkin (GD-LSPG), which uses a nonlinear manifold least-squares Petrov-Galerkin projection to evaluate the temporal evolution of the graph autoencoder's latent representation, and POD latent space dynamics identification (POD-LaSDI), which combines POD-based dimensionality reduction with operator learning. A constraint-enforcement strategy is incorporated into the graph autoencoder to ensure compliance with physical laws. For the studied problems, GNN-LaSDI substantially reduces the computational cost relative to GD-LSPG by eliminating repeated evaluations of the Jacobian of the decoder, while achieving significantly greater predictive accuracy than POD-LaSDI, thereby providing a balance between predictive accuracy and computational efficiency. Additionally, the proposed point cloud error provides a more informative measure of reduced-order model accuracy in capturing the locations of sharp gradients than conventional error metrics.

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Liam K Magargal, Parisa Khodabakhshi. 2026-09-16. Efficient model-order reduction via graph autoencoders and operator learning for systems with sharp gradients using a novel point cloud error metric for performance assessment. https://arxiv.org/abs/2606.23834

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