\documentclass[11pt]{report}
\usepackage{setspace}
\doublespacing
\usepackage[toc,page]{appendix}

\usepackage{pdfpages}
\usepackage[]{amsmath}
\usepackage[]{amsthm}
\usepackage{mathtools}
\numberwithin{equation}{section}
\usepackage[]{amssymb}
\usepackage[margin=1in]{geometry}
\usepackage[]{soul}
\usepackage[]{bbm}
\usepackage[]{cancel}
\usepackage[]{xcolor}
\usepackage[]{enumitem}
\usepackage{mathrsfs}

\usepackage{graphicx}
\usepackage{subfigure}

\usepackage{booktabs}

\usepackage{hyperref}
\hypersetup{
    pdfauthor={Shakil Rafi},
    pdftitle={Dissertation},
    pdfkeywords={neural-networks, stochastic-processes},
    colorlinks = true,
    filecolor = magenta,
    urlcolor = cyan
}


\usepackage[capitalise]{cleveref}
\usepackage{natbib}
\usepackage{neuralnetwork}
\usepackage{witharrows}
\usepackage{stmaryrd}
\usepackage{stackengine,amssymb,graphicx}

\usepackage{listings}

\usepackage{xcolor}

\definecolor{codegreen}{rgb}{0,0.6,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolor}{rgb}{0.95,0.95,0.92}

% Define R language settings
\lstdefinestyle{rstyle}{
  language=R,
  basicstyle=\ttfamily\footnotesize,
  backgroundcolor = \color{backcolor},
  commentstyle=\color{codegreen},
  keywordstyle=\color{magenta},
  numberstyle=\ttfamily\tiny\color{codegray},
  numbers=left,
  stepnumber=1,
  frame=single,
  breaklines=true,
  numbersep = 5pt,
  breakatwhitespace=false,
  tabsize=4,
  morekeywords={install.packages, library, ggplot, aes, geom_bar}
}

\usepackage{fontspec}
\setmonofont{Fira Code}







\usepackage{graphicx}

\DeclareMathAlphabet{\mymathbb}{U}{BOONDOX-ds}{m}{n}

% \usepackage[]{enumerate}
\setlength\parindent{0pt}

\DeclareMathOperator{\Trace}{Trace}
\DeclareMathOperator{\Hess}{Hess}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\domain}{Domain}
\DeclareMathOperator{\lip}{Lip}
\DeclareMathOperator{\var}{Var}
\DeclareMathOperator{\cov}{Cov}
\DeclareMathOperator{\unif}{Unif}
\DeclareMathOperator{\dropout}{Dropout}
\DeclareMathOperator{\bern}{Bern}
\DeclareMathOperator{\norm}{Norm}
\DeclareMathOperator{\diag}{diag}

\usepackage{tikz-cd}

\DeclareMathOperator{\rect}{\mathfrak{r}}
\DeclareMathOperator{\param}{\mathsf{P}}
\DeclareMathOperator{\inn}{\mathsf{I}}
\DeclareMathOperator{\out}{\mathsf{O}}
\DeclareMathOperator{\neu}{\mathsf{NN}}
\DeclareMathOperator{\hid}{\mathsf{H}}
\DeclareMathOperator{\lay}{\mathsf{L}}
\DeclareMathOperator{\dep}{\mathsf{D}}
\DeclareMathOperator{\we}{Weight}
\DeclareMathOperator{\bi}{Bias}
\DeclareMathOperator{\aff}{\mathsf{Aff}}
\DeclareMathOperator{\act}{\mathfrak{a}}
\DeclareMathOperator{\real}{\mathfrak{I}}
\DeclareMathOperator{\id}{\mathsf{Id}}
\DeclareMathOperator{\mult}{Mult}
\DeclareMathOperator{\wid}{\mathsf{W}}
\DeclareMathOperator{\sm}{\mathsf{Sum}}
\DeclareMathOperator{\trn}{Trn}
\DeclareMathOperator{\tun}{\mathsf{Tun}}
\DeclareMathOperator{\cpy}{\mathsf{Cpy}}
\DeclareMathOperator{\ex}{\mathfrak{E}}
\DeclareMathOperator{\lin}{Lin}
\DeclareMathOperator{\relu}{\mathsf{ReLU}}
\DeclareMathOperator{\zero}{Zr}
\DeclareMathOperator{\sqr}{\mathsf{Sqr}}
\DeclareMathOperator{\prd}{\mathsf{Prd}}
\DeclareMathOperator{\pwr}{\mathsf{Pwr}}
\DeclareMathOperator{\xpn}{\mathsf{Xpn}}
\DeclareMathOperator{\rem}{\mathsf{Rem}}
\DeclareMathOperator{\tay}{\mathsf{Tay}}
\DeclareMathOperator{\G}{\mathsf{G}}
\DeclareMathOperator{\F}{\mathsf{F}}
\DeclareMathOperator{\U}{\mathsf{U}}
\DeclareMathOperator{\sP}{\mathsf{P}}
\DeclareMathOperator{\linn}{\mathsf{Lin}}
\DeclareMathOperator{\nrm}{\mathsf{Nrm}}
\DeclareMathOperator{\mxm}{\mathsf{Mxm}}
\DeclareMathOperator{\trp}{\mathsf{Trp}}
\DeclareMathOperator{\etr}{\mathsf{Etr}}
\DeclareMathOperator{\lc}{\left\lceil}
\DeclareMathOperator{\rc}{\right\rceil}
\DeclareMathOperator{\csn}{\mathsf{Csn}}
\DeclareMathOperator{\sne}{\mathsf{Sne}}
\DeclareMathOperator{\pln}{\mathsf{Pln}}
\DeclareMathOperator{\pnm}{\mathsf{Pnm}}
\DeclareMathOperator{\inst}{\mathfrak{I}}
\DeclareMathOperator{\rows}{rows}
\DeclareMathOperator{\columns}{columns}

\DeclareMathOperator{\obj}{obj}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\cod}{cod}

\newcommand{\bbP}{\mathbb{P}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\mft}{\mathfrak{t}}
\newcommand{\f}{\mathfrak{f}}
\newcommand{\C}{\mathfrak{C}}
\newcommand{\n}{\mathscr{N}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\rb}{\right]}
\newcommand{\lb}{\left[}
\newcommand{\lv}{\left|}
\newcommand{\rv}{\right|}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\ve}{\varepsilon}
\newcommand{\les}{\leqslant}
\newcommand{\ges}{\geqslant}



\newcommand\dplus{\ensurestackMath{%
  \stackengine{.5pt}{\Diamond}{\scalebox{.75}[1]{$+$}}{O}{c}{F}{F}{L}}}

\newcommand{\DDiamond}{%
  \begin{tikzpicture}[baseline={(0,-0.2ex)}]
    \draw[rotate=45] (0,0) rectangle (0.2,0.2);
    \draw[rotate=45] (0,0.2) -- (0.2,0);
  \end{tikzpicture}%
}

% Input layer neurons number
\newcommand{\inputnum}{3} 
 
% Hidden layer neurons number
\newcommand{\hiddennum}{5}  
 
% Output layer neurons number
\newcommand{\outputnum}{2} 

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{claim}[theorem]{Claim}