🤖 AI Summary
Traditional linear model reduction techniques, such as Proper Orthogonal Decomposition (POD), struggle to effectively handle high-dimensional dynamical systems featuring sharp gradients. This work proposes the GNN-LaSDI framework, which integrates a graph autoencoder for nonlinear dimensionality reduction and employs operator learning to directly model temporal evolution in the latent space. The approach accurately captures the locations of steep gradients while maintaining computational efficiency. Furthermore, it introduces a novel point-cloud–oriented error metric that provides a more intuitive assessment of local accuracy. In two representative numerical experiments, GNN-LaSDI achieves significantly higher accuracy than POD-LaSDI and substantially reduces computational cost compared to GD-LSPG, thereby striking an effective balance between accuracy and efficiency.
📝 Abstract
This study investigates the efficient deployment of graph autoencoders, a class of graph neural networks (GNNs), for model-order reduction (MOR) of high-dimensional dynamical systems. The proposed framework leverages graph autoencoders to perform nonlinear dimensionality reduction, enabling low-dimensional representations of systems characterized by sharp gradients for which conventional linear approximations, such as proper orthogonal decomposition (POD), are inadequate. Specifically, this study introduces graph neural network latent space dynamics identification (GNN-LaSDI). GNN-LaSDI employs an operator learning framework to directly approximate the temporal evolution of the graph autoencoder's latent representation. The performance of GNN-LaSDI is assessed against both geometric deep least-squares Petrov-Galerkin (GD-LSPG and POD latent space dynamics identification (POD-LaSDI), which combines POD-based dimensionality reduction with operator learning. In addition to standard error metrics, this work presents a novel point cloud error metric specifically tailored to evaluate the accuracy of the identified locations of sharp gradients within the solution. The effectiveness of the metric and the proposed MOR framework is demonstrated through two numerical experiments featuring sharp gradients. For the studied problems, GNN-LaSDI incurs a substantially lower computational cost than GD-LSPG, though it remains slightly more computationally expensive than POD-LaSDI. However, GNN-LaSDI achieves significantly greater accuracy than POD-LaSDI, thereby providing a balance between predictive accuracy and computational speedup. Additionally, the results indicate that the proposed point cloud error provides a more intuitive and informative measure of reduced-order model accuracy in regions with sharp gradients than conventional error metrics.