Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lorenzo Tomada, Federico Pichi, Gianluigi Rozza

TL;DR

This work introduces 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.

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

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

TL;DR

This work introduces 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.

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
Paper Structure (25 sections, 2 theorems, 31 equations, 14 figures, 3 tables, 1 algorithm)

This paper contains 25 sections, 2 theorems, 31 equations, 14 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Methodology
  4. Formal problem statement
  5. Brief introduction to graph neural networks
  6. Latent Dynamics Networks
  7. Latent Dynamics Graph Convolutional Networks
  8. Theoretical guarantees
  9. Interpolation of latent trajectories
  10. Numerical results
  11. Advection-diffusion equation
  12. Problem definition and data collection
  13. Square advection ()
  14. Moving hole advection ()
  15. Analysis and interpolation of the latent states
  16. ...and 10 more sections

Key Result

Corollary 2.6

Let $I_h$ be a discretization of $I$ with step size $\Delta t$. Assume that the initial condition of eq:abstract is fixed and equal to $\bm u_h(t_0)$ for all $\bm\mu\in\mathbb{P}$. If the perfect embedding assumption holds, then for each $\varepsilon >0$, there exist: such that with where $\{(\omega_j, t_j) \,|\, j=0,\dots, k \}$ are the quadrature weights and nodes.

Figures (14)

  • Figure 1: Schematic representation of the architecture.
  • Figure 2: Relative errors for all test trajectories and latent evolution w.r.t. time.
  • Figure 3: Plot of the simulated solution (left), the full-order one (middle) and relative $L^2$ error field (right) for $\boldsymbol{\mu}=(1, 1)$ at time $t=2$.
  • Figure 4: Relative errors for all test trajectories w.r.t. time.
  • Figure 5: Plot of the simulated solution (left), the full-order one (middle) and relative $L^2$ error (right) for two different values of $\boldsymbol{\mu}\in\mathbb{P}_h$.
  • ...and 9 more figures

Theorems & Definitions (13)

  • Remark 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4: Scaling
  • Definition 2.5
  • Corollary 2.6: UAT for encoder-free architectures
  • proof
  • Remark 2.7
  • Proposition 2.8
  • Remark 2.9
  • ...and 3 more