\include{preamble}
\include{commands}

\title{Artificial  Neural Networks Applied to Stochastic Monte Carlo as a Way to Approximate Modified Heat Equations, and Their Associated Parameters, Depths, and Accuracies.}
\author{Shakil Rafi}
\begin{document}
\maketitle  

\tableofcontents

\part{On Convergence of Brownian Motion Monte Carlo}

\include{Introduction}

\include{Brownian_motion_monte_carlo}

\include{u_visc_sol}

\include{brownian_motion_monte_carlo_non_linear_case}

\part{A Structural Description of Artificial Neural Networks}

\include{neural_network_introduction}

\include{ann_product}

%\include{modified_mlp_associated_nn}

\include{ann_first_approximations}

\part{A deep-learning solution for $u$ and Brownian motions}

\include{ann_rep_brownian_motion_monte_carlo}

\include{conclusions-further-research}

\nocite{*}
\singlespacing
\bibliography{main.bib}
\bibliographystyle{apa}

\include{appendices}

\end{document}