Wasserstein Data Generation Models

Model-based or PDE-based uncertainty quantification tries to reconstruct the probability distribution of the quantity of interest, represented for example by the solution of a parametric PDE, given the probability distribution of the uncertainty on the model data. Typically this approach requires the use of Monte-Carlo methods and the consequential construction of very large ensemble, each member of which corresponds to the solution of a PDE with data sampled from their uncertainty distribution. This constitutes an enormous task often unfeasible in terms of computational time. It is then very important to be able to reduce the computational requirements of the construction of the Monte-Carlo ensemble. The main focus of this thesis deals with the efficient construction of these Monte-Carlo ensembles for general parametric scalar PDEs by leveraging on the strength of existing AI tools to ascertain if they can be used effectively for the purpose of Monte-Carlo data generation. In particular, we consider the weaker requirement of achieving accurate reproduction not of every single realization of the ensemble but only of its empirical moments. Specifically, we work with Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs)m because of the possibility to enforce on the trained latent space a given probability distribution. A well trained VAE will then allow the construction of an accurate ensemble capable of reproducing the empirical moments of the sought distribution by sampling from the pre-imposed probability distribution of the latent space and using the decoder to generate the full dimensional data. The success of these generative models depends fundamentally by their accurate training, which require the evaluation of the distance between two parametric PDE solution. Standard implementations employ the Jensen-Shannon or Kullback-Leibler divergences for this task. However, the limited continuity and accuracy of these distances are believed to be the main cause of inaccuracy in the training phase. To alleviate this problem, the Wasserstein distance, a metric used to measure the distance between probability distributions, has been often employed. However, standard approaches, because of limitations due essentially to the computational difficulties in the calculation of the Wasserstein distance, admit a tradeoff in favor efficiency sacrificing accuracy. In the case of data generation this sacrifice cannot be done and accurate evaluation of the Wasserstein distance seems mandatory. The work in this thesis builds on this idea by employing a modern and recently developed Wasserstein distance calculation capable of accuractely calculate the Wasserstein-1 distance between two distributions. Thus, in the first part of the thesis builds on this work with the aim of adapting this approach towards its implementation in VAEs or GANs. This work is still in progress. In the second part, we address the problem of implementing this distance calculator within the framework of Uncertainty Quantification (UQ) and parametric Partial Differential Equations (PDEs). The primary goal is to quantify the effect on the solution of the uncertainties in the input parameters. We employ Monte Carlo (MC) method to evaluate the quantities of interest, where we utilize data generation techniques for the construction of MC ensemble. We present the current results of our implementation of the standard VAE for this problem, along with some initial results of the Wasserstein Autoencoder (WAE) model. Lastly, we outline our future directions to achieve the ultimate goal of this research.