Geometric and functional representations of stochastic neural dynamical systems: from realization theory to controlled approximation

Veeravalli, Tanya

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https://hdl.handle.net/2142/129887 Copy

Description

Title

Geometric and functional representations of stochastic neural dynamical systems: from realization theory to controlled approximation

Author(s)

Veeravalli, Tanya

Issue Date

2025-07-18

Director of Research (if dissertation) or Advisor (if thesis)

Raginsky, Maxim

Doctoral Committee Chair(s)

Raginsky, Maxim

Committee Member(s)

• Srikant, Rayadurgam
• Belabbas, Mohamed Ali
• Zhao, Zhizhen

Department of Study

Electrical & Computer Eng

Discipline

Electrical & Computer Engr

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• optimal control
• neural dynamical systems
• stochastic realization theory
• nonlinear controllability
• function approximation

Abstract

There has been a great deal of interest in understanding continuous-time processes in deep learning, in particular methods related to (stochastic) control for improving diffusion models. In this work, we explore various facets of function approximation and realization problems through the lens of dynamical systems theory, neural stochastic differential equations (neural SDEs), and differential geometry. A neural SDE is an Itô diffusion process whose drift and diffusion matrices are elements of some parametric families. We cover topics from estimating the transition density of both uniformly elliptic and possibly degenerate diffusion processes by leveraging tools from sub-Riemannian geometry and stochastic control. There are many nuanced insights we can get into the behavior of deep neural networks and diffusion models by studying properties of associated problems in optimal control theory and drawing on other tools from the rich mathematical physics literature. The geometric insights explain the underlying noise structure and controllability properties of a stochastic dynamical system while also explaining the expressive power of the stochastic system.

Graduation Semester

2025-08

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/129887

Copyright and License Information

Copyright 2025 Tanya Veeravalli

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