Generative adversarial reduced order modelling

Abstract

In this work, we present GAROM, a new approach for reduced order modeling (ROM) based on generative adversarial networks (GANs). GANs attempt to learn to generate data with the same statistics of the underlying distribution of a dataset, using two neural networks, namely discriminator and generator. 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, introducing a data-driven generative adversarial model able to learn solutions to parametric differential equations. In the presented methodology, the discriminator is modeled as an autoencoder, extracting relevant features of the input, and a conditioning mechanism is applied 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 generalization, and perform a convergence study of the method.

Figure 1

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 2

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 3

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.

Figure 4

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.