Abstract:
Gradient flow methods have been classical tools for analyzing and numerically solving partial differential equations, traditionally formulated in Hilbert spaces. This thesis extends that framework to gradient flows in metric spaces and explores both forward problems, which are concerned with accurately solving evolution equations while preserving stability, and inverse, data-driven problems that aim to uncover underlying dynamics through the lens of transport maps.
The first part of the thesis focuses on high-order numerical schemes for gradient flows in metric spaces. We introduce new criteria for ensuring energy stability in multi-step, multi-stage, and mixed numerical schemes, specifically designed for evolution equations that arise as gradient flows with respect to a metric. These criteria lead to the construction of second- and third- order accurate, energy-stable schemes. We validate the order of accuracy and stability of these schemes on several partial differential equations formulated as gradient flows with respect to the quadratic Wasserstein metric, including the heat, Fokker-Planck, and porous medium equations.
The second part of the thesis addresses the inverse problem of identifying governing Fokker-Planck equations from time-evolving data by exploiting the underlying gradient flow structure and suitable transport maps. We develop a framework that simultaneously reconstructs system dynamics and evolving probability density functions using Knothe-Rosenblatt rearrangements and the Kullback-Leibler divergence. The model is implemented using machine learning, with both the system dynamics and the structure of the Knothe-Rosenblatt rearrangements represented by neural networks. We further introduce an extension to Bayesian modeling, grounded in variational inference via the evidence lower bound (ELBO), which enables uncertainty quantification. The choice of likelihood functions is discussed within the context of this framework. The approach is demonstrated through numerical experiments in spaces of up to five dimensions.
Gradient flow methods have been classical tools for analyzing and numerically solving partial differential equations, traditionally formulated in Hilbert spaces. This thesis extends that framework to gradient flows in metric spaces and explores both forward problems, which are concerned with accurately solving evolution equations while preserving stability, and inverse, data-driven problems that aim to uncover underlying dynamics through the lens of transport maps.
The first part of the thesis focuses on high-order numerical schemes for gradient flows in metric spaces. We introduce new criteria for ensuring energy stability in multi-step, multi-stage, and mixed numerical schemes, specifically designed for evolution equations that arise as gradient flows with respect to a metric. These criteria lead to the construction of second- and third- order accurate, energy-stable schemes. We validate the order of accuracy and stability of these schemes on several partial differential equations formulated as gradient flows with respect to the quadratic Wasserstein metric, including the heat, Fokker-Planck, and porous medium equations.
The second part of the thesis addresses the inverse problem of identifying governing Fokker-Planck equations from time-evolving data by exploiting the underlying gradient flow structure and suitable transport maps. We develop a framework that simultaneously reconstructs system dynamics and evolving probability density functions using Knothe-Rosenblatt rearrangements and the Kullback-Leibler divergence. The model is implemented using machine learning, with both the system dynamics and the structure of the Knothe-Rosenblatt rearrangements represented by neural networks. We further introduce an extension to Bayesian modeling, grounded in variational inference via the evidence lower bound (ELBO), which enables uncertainty quantification. The choice of likelihood functions is discussed within the context of this framework. The approach is demonstrated through numerical experiments in spaces of up to five dimensions.
| Building: | West Hall |
|---|---|
| Event Type: | Presentation |
| Tags: | Dissertation, Graduate, Graduate Students, Mathematics |
| Source: | Happening @ Michigan from Dissertation Defense - Department of Mathematics, Department of Mathematics |
