Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs

TL;DR

This paper introduces LD-GCN, a graph neural network architecture for model order reduction of parameterized PDEs that learns interpretable low-dimensional dynamics and enables accurate, geometry-aware time extrapolation.

Contribution

The work presents a novel encoder-free GNN framework that models latent dynamics directly, validated theoretically and demonstrated on complex mechanics problems with geometric parameters.

Findings

Effective low-dimensional representation of dynamical systems.

Supports zero-shot prediction via latent space interpolation.

Successfully applied to Navier-Stokes bifurcation detection.

Abstract

Graph Neural Networks (GNNs) are emerging as powerful tools for nonlinear Model Order Reduction (MOR) of time-dependent parameterized Partial Differential Equations (PDEs). However, existing methodologies struggle to combine geometric inductive biases with interpretable latent behavior, overlooking dynamics-driven features or disregarding spatial information. In this work, we address this gap by introducing Latent Dynamics Graph Convolutional Network (LD-GCN), a purely data-driven, encoder-free architecture that learns a global, low-dimensional representation of dynamical systems conditioned on external inputs and parameters. The temporal evolution is modeled in the latent space and advanced through time-stepping, allowing for time-extrapolation, and the trajectories are consistently decoded onto geometrically parameterized domains using a GNN. Our framework enhances interpretability by enabling the analysis of the reduced dynamics and supporting zero-shot prediction through latent interpolation. The methodology is mathematically validated via a universal approximation theorem for encoder-free architectures, and numerically tested on complex computational mechanics problems involving physical and geometric parameters, including the detection of bifurcating phenomena for Navier-Stokes equations. Code availability: https://github.com/lorenzotomada/ld-gcn-rom