Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generative Adversarial Reduced Order Modelling

Dario Coscia, Nicola Demo, Gianluigi Rozza

TL;DR

GAROM presents a generative adversarial reduced order modeling framework for parametric PDEs by integrating a generator $G(\mathbf{z}|\mathbf{c})$ with a discriminative autoencoder conditioned on parameters $\mathbf{c}$, enabling real-time inference and probabilistic predictions. A regularized variant, r-GAROM, enforces solution uniqueness and improves stability. The approach is validated on Gaussian, Graetz, and Lid Cavity benchmarks, where it often surpasses traditional ROMs in generalization while providing principled uncertainty estimates via Monte Carlo sampling and simple bound-based UQ strategies. The work suggests promising extensions to time-dependent problems and continuous, discretization-invariant mappings, underscoring GAROM's potential for scalable, uncertainty-aware ROM in complex CFD contexts.

Abstract

In this work, we present GAROM, a new approach for reduced order modelling (ROM) based on generative adversarial networks (GANs). GANs have the potential to learn data distribution and generate more realistic data. While widely applied in many areas of deep learning, little research is done on their application for ROM, i.e. approximating a high-fidelity model with a simpler one. In this work, we combine the GAN and ROM framework, by introducing a data-driven generative adversarial model able to learn solutions to parametric differential equations. The latter is achieved by modelling the discriminator network as an autoencoder, extracting relevant features of the input, and applying a conditioning mechanism to the generator and discriminator networks specifying the differential equation parameters. We show how to apply our methodology for inference, provide experimental evidence of the model generalisation, and perform a convergence study of the method.

Generative Adversarial Reduced Order Modelling

TL;DR

GAROM presents a generative adversarial reduced order modeling framework for parametric PDEs by integrating a generator with a discriminative autoencoder conditioned on parameters , enabling real-time inference and probabilistic predictions. A regularized variant, r-GAROM, enforces solution uniqueness and improves stability. The approach is validated on Gaussian, Graetz, and Lid Cavity benchmarks, where it often surpasses traditional ROMs in generalization while providing principled uncertainty estimates via Monte Carlo sampling and simple bound-based UQ strategies. The work suggests promising extensions to time-dependent problems and continuous, discretization-invariant mappings, underscoring GAROM's potential for scalable, uncertainty-aware ROM in complex CFD contexts.

Abstract

In this work, we present GAROM, a new approach for reduced order modelling (ROM) based on generative adversarial networks (GANs). GANs have the potential to learn data distribution and generate more realistic data. While widely applied in many areas of deep learning, little research is done on their application for ROM, i.e. approximating a high-fidelity model with a simpler one. In this work, we combine the GAN and ROM framework, by introducing a data-driven generative adversarial model able to learn solutions to parametric differential equations. The latter is achieved by modelling the discriminator network as an autoencoder, extracting relevant features of the input, and applying a conditioning mechanism to the generator and discriminator networks specifying the differential equation parameters. We show how to apply our methodology for inference, provide experimental evidence of the model generalisation, and perform a convergence study of the method.
Paper Structure (19 sections, 13 equations, 4 figures, 1 table)

This paper contains 19 sections, 13 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Conditional Boundary Equilibrium GAN
  4. GAROM implementation details
  5. GAROM predictive distribution
  6. Uncertainty quantification strategies
  7. Datasets description
  8. Parametric Gaussian
  9. Graetz Problem
  10. Lid Cavity
  11. Baseline models
  12. Software details
  13. Training setup and model architecture
  14. Results
  15. Model inference
  16. ...and 4 more sections

Figures (4)

  • Figure 1: A schematic representation for GAROM generator and discriminator. The Generator input is the concatenation of random noise $\mathbf{z}$, and the conditioning representation $f_\tau(\mathbf{c})$. The Discriminator encodes the input obtaining a latent vector, which is concatenated with the conditioning representation $g_\psi(\mathbf{c})$ before it is passed to the decoder.
  • Figure 2: r-GAROM inference results. The images show the generated snapshot representing the magnitude of the unknown field for a testing parameter using a latent dimension of $64$, compared to the corresponding high-fidelity solution. Top: Gaussian dataset. Center: Graetz dataset. Bottom: Lid cavity dataset.
  • Figure 3: Distribution of the $\delta$ difference. The graph depicts, for each latent dimension, train (red) and test (blue) distribution of the $\delta$. Top: r-GAROM model. Bottom: GAROM model.
  • Figure 4: r-GAROM convergence graph in $l_2$ relative error for multiple training. The solid line indicates the average across all simulations. The shaded area represents the interval obtained by taking the maximum and minimum error across all simulations. The total number of training is $5$.