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\include { preamble}
\include { commands}
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\title { Artificial Neural Networks Applied to Stochastic Monte Carlo as a Way to Approximate Modified Heat Equations, and Their Associated Parameters, Depths, and Accuracies.}
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\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}
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%\include{modified_mlp_associated_nn}
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\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}