Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition

TL;DR

This paper introduces a novel reduced-order modeling approach combining graph convolutional autoencoders, tensor train decomposition, and operator inference to improve temporal extrapolation in complex dynamical systems.

Contribution

It develops a time-consistent reduced-order model integrating TT decomposition and OpInf, and enhances generalization with a multi-fidelity DeepONet framework for complex geometries.

Findings

Effective extrapolation in heat-conduction, advection-diffusion, vortex-shedding cases

Outperforms state-of-the-art methods like MeshGraphNets

Demonstrates robustness on unstructured grids and complex geometries

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.