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Learning Stochastic Governing Laws from Noisy Data Using Normalizing Flows
Title
Learning Stochastic Governing Laws from Noisy Data Using Normalizing Flows
Creator
McClure, William Jacob
Date
2021
Description
With the increasing availability of massive collections of data, researchers in all sciences need tools to synthesize useful and pertinent...
Show moreWith the increasing availability of massive collections of data, researchers in all sciences need tools to synthesize useful and pertinent descriptors of the systems they study. Perhaps the most fundamental knowledge of a dynamical system is its governing laws, which describe its evolution through time and can be lever-aged for a number of analyses about its behavior. We present a novel technique for learning the infinitesimal generator of a Markovian stochastic process from large, noisy datasets generated by a stochastic system. Knowledge of the generator in turn allows us to find the governing laws for the process. This technique relies on normalizing flows, neural networks that estimate probability densities, to learn the density of time-dependent stochastic processes. We establish the efficacy of this technique on multiple systems with Brownian noise, and use our learned governing laws to perform analysis on one system by solving for its mean exit time. Our approach also allows us to learn other dynamical behaviors such as escape probability and most probable pathways in a system. The potential impact of this technique is far-reaching, since most stochastic processes in various fields are assumed to be Markovian, and the only restriction for applying our method is available data from a time near the beginning of an experiment or recording.
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NUMERICAL ANALYSIS ON MAXIMUM LIKELIHOOD ESTIMATORS FOR LINEAR PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Title
NUMERICAL ANALYSIS ON MAXIMUM LIKELIHOOD ESTIMATORS FOR LINEAR PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Creator
Zhang, Jun
Date
2019
Description
The thesis contributes to the numerical analysis on statistical inference for stochastic partial differential equations (SPDEs). We study the...
Show moreThe thesis contributes to the numerical analysis on statistical inference for stochastic partial differential equations (SPDEs). We study the maximum likelihood estimation problem of the drift parameter for a large class of linear parabolic SPDEs. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that one path of the first N Fourier modes is observed continuously in a fixed finite time interval [0, T]. We first provide a review of the asymptotic properties of the maximum likelihood estimator (MLE) of the drift parameter in the large number of Fourier modes regime, N  ∞, while the time horizon T > 0 is fixed. The main part of this thesis is dedicated to the numerical study of the asymptotic properties of the MLEs for two examples of linear parabolic SPDEs: the one-dimensional stochastic heat equation and a d-dimensional linear, diagonalizable, parabolic SPDE, where d  ℕ. For the one-dimensional stochastic heat equation, we perform the sensitivity analysis to assess the effect of changes in model parameters on the speed of convergence of the MLE. For the second linear parabolic SPDE, our simulations verify the theoretical results in the literature that both the consistency and asymptotic normality of the MLE hold for such equation only when d ≥ 2.
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Choice-Distinguishing Colorings of Cartesian Products of Graphs
Title
Choice-Distinguishing Colorings of Cartesian Products of Graphs
Creator
Tomlins, Christian James
Date
2022
Description
A coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every...
Show moreA coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every vertex color. The distinguishing number, $D(G)$, of a graph $G$ is the smallest positive integer $k$ such that there exists a distinguishing coloring $f: V(G)\rightarrow [k]$ and was introduced by Albertson and Collins in their paper ``Symmetry Breaking in Graphs.'' By restricting what kinds of colorings are considered, many variations of distinguishing numbers have been studied. In this paper, we study proper list-colorings of graphs which are also distinguishing and investigate the choice-distinguishing number $\text{ch}_D(G)$ of a graph $G$. Primarily, we focus on the choice-distinguishing number of Cartesian products of graphs. We determine the exact value of $\text{ch}_D(G)$ for lattice graphs and prism graphs and provide an upper bound on the choice-distinguishing number of the Cartesian products of two relatively prime graphs, assuming a sufficient condition is satisfied. We use this result to bound the choice distinguishing number of toroidal grids and the Cartesian product of a tree with a clique. We conclude with a discussion on how, depending on the graphs $G$ and $H$, we may weaken the sufficient condition needed to bound $\text{ch}_D(G\square H)$.
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