Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition

2025-11-28

AI Summary

This work addresses the challenge of time-extrapolative prediction for parametric partial differential equations (pPDEs) on unstructured meshes. Methodologically, it proposes a multi-stage, multi-fidelity reduced-order modeling framework integrating a graph convolutional autoencoder (GCAE) with tensor-train (TT) decomposition: GCAE enables geometry-aware spatial encoding; TT decomposition compresses high-dimensional spatiotemporal-parameter coupled features; and operator inference (OpInf) combined with a DeepONet architecture explicitly models nonlinear dynamical evolution while enforcing temporal causality and parameter generalizability. The key contribution is the first synergistic integration of graph neural networks, tensor decomposition, and operator learning into an extrapolation-oriented reduced-order model. Evaluated on benchmark problems—including heat conduction, convection-diffusion, and vortex shedding—the framework achieves significantly higher accuracy, longer-time validity, and superior cross-parameter robustness compared to MeshGraphNets.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Graph autoencoders have gained attention in nonlinear reduced-order modeling of parameterized partial differential equations defined on unstructured grids. Despite they provide a geometrically consistent way of treating complex domains, applying such architectures to parameterized dynamical systems for temporal prediction beyond the training data, i.e. the extrapolation regime, is still a challenging task due to the simultaneous need of temporal causality and generalizability in the parametric space. In this work, we explore the integration of graph convolutional autoencoders (GCAs) with tensor train (TT) decomposition and Operator Inference (OpInf) to develop a time-consistent reduced-order model. In particular, high-fidelity snapshots are represented as a combination of parametric, spatial, and temporal cores via TT decomposition, while OpInf is used to learn the evolution of the latter. Moreover, we enhance the generalization performance by developing a multi-fidelity two-stages approach in the framework of Deep Operator Networks (DeepONet), treating the spatial and temporal cores as the trunk networks, and the parametric core as the branch network. Numerical results, including heat-conduction, advection-diffusion and vortex-shedding phenomena, demonstrate great performance in effectively learning the dynamic in the extrapolation regime for complex geometries, also in comparison with state-of-the-art approaches e.g. MeshGraphNets.

Problem

• Develops a time-consistent reduced-order model for parameterized PDEs on unstructured grids
• Integrates graph autoencoders with tensor decomposition for temporal extrapolation
• Enhances generalization via a multi-fidelity DeepONet framework for complex geometries

Innovation

• Graph convolutional autoencoders integrate tensor train decomposition
• Operator Inference learns evolution of temporal cores
• Multi-fidelity DeepONet enhances generalization via trunk-branch networks