Learning Dynamics from Data Using Optimal Transport Techniques and Applications

Ma, Shaojun

Title:

Learning Dynamics from Data Using Optimal Transport Techniques and Applications

dc.contributor.advisor	Zhou, Haomin
dc.contributor.author	Ma, Shaojun
dc.contributor.committeeMember	Zha, Hongyuan
dc.contributor.committeeMember	Ye, Xiaojing
dc.contributor.committeeMember	Li, Wuchen
dc.contributor.committeeMember	Tao, Molei
dc.contributor.committeeMember	Dieci, Luca
dc.contributor.department	Mathematics
dc.date.accessioned	2022-08-25T13:34:33Z
dc.date.available	2022-08-25T13:34:33Z
dc.date.created	2022-08
dc.date.issued	2022-07-08
dc.date.submitted	August 2022
dc.date.updated	2022-08-25T13:34:33Z
dc.description.abstract	Optimal Transport has been studied widely in recent years, the concept of Wasserstein distance brings a lot of applications in computational mathematics, machine learning, engineering, even finance areas. Meanwhile, people are gradually realizing that as the amount of data as well as the needs of utilizing data increase vastly, data-driven models have great potentials in real-world applications. In this thesis, we apply the theories of OT and design data-driven algorithms to form and compute various OT problems. We also build a framework to learn inverse OT problem. Furthermore, we develop OT and deep learning based models to solve stochastic differential equations, optimal control, mean field games related problems, all in data-driven settings. In Chapter 2, we provide necessary mathematical concepts and results that form the basis of this thesis. It contains brief surveys of optimal transport, stochastic differential equations, Fokker-Planck equations, deep learning, optimal controls and mean field games. Chapter 3 to Chapter 5 present several scalable algorithms to handle optimal transport problems within different settings. Specifically, Chapter 3 shows a new saddle scheme and learning strategy for computing the Wasserstein geodesic, as well as the Wasserstein distance and OT map between two probability distributions in high dimensions. We parametrize the map and Lagrange multipliers as neural networks. We demonstrate the performance of our algorithms through a series of experiments with both synthetic and realistic data. Chapter 4 presents a scalable algorithm for computing the Monge map between two probability distributions since computing the Monge maps remains challenging, in spite of the rapid developments of the numerical methods for optimal transport problems. Similarly, we formulate the problem as a mini-max problem and solve it via deep learning. The performance of our algorithms is demonstrated through a series of experiments with both synthetic and realistic data. In Chapter 5 we study OT problem in an inverse view, which we also call Inverse OT (IOT) problem. IOT also refers to the problem of learning the cost function for OT from observed transport plan or its samples. We derive an unconstrained convex optimization formulation of the inverse OT problem. We provide a comprehensive characterization of the properties of inverse OT, including uniqueness of solutions. We also develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for discrete OT, and the other one is a learning based algorithm that parameterizes the cost function as a deep neural network for continuous OT. Our numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state-of-the-art methods. In Chapter 6 we propose a novel method using the weak form of Fokker Planck Equation (FPE) --- a partial differential equation --- to describe the density evolution of data in a sampled form, which is then combined with Wasserstein generative adversarial network (WGAN) in the training process. In such a sample-based framework we are able to learn the nonlinear dynamics from aggregate data without explicitly solving FPE. We demonstrate our approach in the context of a series of synthetic and real-world data sets. Chapter 7 introduces the application of OT and neural networks in optimal density control. Particularly, we parametrize the control strategy via neural networks, and provide an algorithm to learn the strategy that can drive samples following one distribution to new locations following target distribution. We demonstrate our method in both synthetic and realistic experiments, where we also consider perturbation fields. Finally Chapter 8 presents applications of mean field game in generative modeling and finance area. With more details, we build a GAN framework upon mean field game to generate desired distribution starting with white noise, we also investigate its connection to OT. Moreover, we apply mean field game theories to study the equilibrium trading price in stock markets, we demonstrate the theoretical result by conducting experiments on real trading data.
dc.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/67233
dc.language.iso	en_US
dc.publisher	Georgia Institute of Technology
dc.subject	Optimal Transport
dc.subject	Deep Learning
dc.subject	Fokker Planck Equation
dc.subject	Stochastic Differential Equation
dc.subject	Hamiltonian System
dc.subject	Optimal Control
dc.subject	Mean Field Game
dc.title	Learning Dynamics from Data Using Optimal Transport Techniques and Applications
dc.type	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.advisor	Zhou, Haomin
local.contributor.corporatename	College of Sciences
local.contributor.corporatename	School of Mathematics
relation.isAdvisorOfPublication	6289877f-beee-44f1-88b4-761e90d959e7
relation.isOrgUnitOfPublication	85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication	84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level	Doctoral