## Citation

Zhang, Shumao
(2023)
*Singularity Formation in the High-Dimensional Euler Equations and Sampling of High-Dimensional Distributions by Deep Generative Networks.*
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/8had-3a90.
https://resolver.caltech.edu/CaltechTHESIS:09202022-034157716

## Abstract

High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ℝⁿ, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality.

In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is C^{α} Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α < α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor ε as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength ε < ε*, and this upper bound ε* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if ε < 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up.

In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.

Item Type: | Thesis (Dissertation (Ph.D.)) | |||||||||
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Subject Keywords: | Fluid Dynamics; Euler Equations; Singularity Formation; Deep Generative Networks; Bayesian Inverse Problems; Image Synthesis | |||||||||

Degree Grantor: | California Institute of Technology | |||||||||

Division: | Engineering and Applied Science | |||||||||

Major Option: | Applied And Computational Mathematics | |||||||||

Thesis Availability: | Public (worldwide access) | |||||||||

Research Advisor(s): | - Hou, Thomas Y.
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Thesis Committee: | - Meiron, Daniel I. (chair)
- Bhattacharya, Kaushik
- Hou, Thomas Y.
- Owhadi, Houman
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Defense Date: | 23 August 2022 | |||||||||

Record Number: | CaltechTHESIS:09202022-034157716 | |||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:09202022-034157716 | |||||||||

DOI: | 10.7907/8had-3a90 | |||||||||

Related URLs: |
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ORCID: |
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Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 15033 | |||||||||

Collection: | CaltechTHESIS | |||||||||

Deposited By: | Shumao Zhang | |||||||||

Deposited On: | 25 Oct 2022 21:47 | |||||||||

Last Modified: | 25 Apr 2023 15:18 |

## Thesis Files

PDF (Full thesis)
- Final Version
See Usage Policy. 42MB |

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