Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition

TL;DR

This work tackles time extrapolation for parameterized, nonlinear PDEs on unstructured meshes by marrying Graph Convolutional Autoencoders with Tensor Train decomposition and Operator Inference to form a time-consistent reduced-order model (LF-TTOI). It then extends this framework with a multi-fidelity DeepONet (MF-TTOI-DeepONet) to further boost accuracy by learning residual corrections in a two-stage offline training regime. Across heat conduction, advection-diffusion, and Navier–Stokes with vortex shedding, the proposed LF-TTOI and especially MF-TTOI-DeepONet demonstrate strong extrapolation capabilities and data efficiency, often outperforming POD-based and MeshGraphNets approaches in scarce-data regimes. The results highlight the value of decoupling parametric and temporal cores via TT and enriching temporal dynamics with OpInf and DeepONet-inspired residual learning for robust, geometry-aware ROMs.

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.

Figures (20)

• Figure 1: Visual representation of the GCA-ROM.
• Figure 2: The framework of LF-TTOI.
• Figure 3: The framework of MF-TTOI-DeepONet.
• Figure 4: Geometrical setup of the heat conductivity problem.
• Figure 5: Error ${\bar{\varepsilon}}_{\text{GCA-ROM}}(t)$ of the plain GCA-ROM on test parameter set.