69 lines
6.7 KiB
TeX
69 lines
6.7 KiB
TeX
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\contentsline {part}{I\hspace {1em}On Convergence of Brownian Motion Monte Carlo}{4}{part.1}%
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\contentsline {chapter}{\numberline {1}Introduction.}{5}{chapter.1}%
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\contentsline {section}{\numberline {1.1}Motivation}{5}{section.1.1}%
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\contentsline {section}{\numberline {1.2}Notation, Definitions \& Basic notions.}{6}{section.1.2}%
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\contentsline {subsection}{\numberline {1.2.1}Norms and Inner Product}{7}{subsection.1.2.1}%
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\contentsline {subsection}{\numberline {1.2.2}Probability Space and Brownian Motion}{8}{subsection.1.2.2}%
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\contentsline {subsection}{\numberline {1.2.3}Lipschitz and Related Notions}{10}{subsection.1.2.3}%
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\contentsline {subsection}{\numberline {1.2.4}Kolmogorov Equations}{12}{subsection.1.2.4}%
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\contentsline {subsection}{\numberline {1.2.5}Linear Algebra Notation and Definitions}{13}{subsection.1.2.5}%
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\contentsline {subsection}{\numberline {1.2.6}$O$-type Notation and Function Growth}{15}{subsection.1.2.6}%
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\contentsline {subsection}{\numberline {1.2.7}The Concatenation of Vectors \& Functions}{16}{subsection.1.2.7}%
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\contentsline {chapter}{\numberline {2}Brownian Motion Monte Carlo}{19}{chapter.2}%
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\contentsline {section}{\numberline {2.1}Brownian Motion Preliminaries}{19}{section.2.1}%
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\contentsline {section}{\numberline {2.2}Monte Carlo Approximations}{25}{section.2.2}%
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\contentsline {section}{\numberline {2.3}Bounds and Covnvergence}{26}{section.2.3}%
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\contentsline {chapter}{\numberline {3}That $u$ is a Viscosity Solution}{35}{chapter.3}%
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\contentsline {section}{\numberline {3.1}Some Preliminaries}{35}{section.3.1}%
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\contentsline {section}{\numberline {3.2}Viscosity Solutions}{39}{section.3.2}%
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\contentsline {section}{\numberline {3.3}Solutions, Characterization, and Computational Bounds to the Kolmogorov Backward Equations}{58}{section.3.3}%
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\contentsline {chapter}{\numberline {4}Brownian motion Monte Carlo of the non-linear case}{64}{chapter.4}%
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\contentsline {part}{II\hspace {1em}A Structural Description of Artificial Neural Networks}{66}{part.2}%
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\contentsline {chapter}{\numberline {5}Introduction and Basic Notions About Neural Networks}{67}{chapter.5}%
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\contentsline {section}{\numberline {5.1}The Basic Definition of ANNs and realizations of ANNs}{67}{section.5.1}%
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\contentsline {section}{\numberline {5.2}Compositions of ANNs}{70}{section.5.2}%
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\contentsline {subsection}{\numberline {5.2.1}Composition}{71}{subsection.5.2.1}%
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\contentsline {section}{\numberline {5.3}Parallelization of ANNs of Equal Depth}{76}{section.5.3}%
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\contentsline {section}{\numberline {5.4}Parallelization of ANNs of Unequal Depth}{80}{section.5.4}%
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\contentsline {section}{\numberline {5.5}Affine Linear Transformations as ANNs and Their Properties.}{82}{section.5.5}%
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\contentsline {section}{\numberline {5.6}Sums of ANNs of Same End-widths}{84}{section.5.6}%
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\contentsline {subsection}{\numberline {5.6.1}Neural Network Sum Properties}{85}{subsection.5.6.1}%
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\contentsline {subsection}{\numberline {5.6.2}Sum of ANNs of Unequal Depth But Same End-widths}{92}{subsection.5.6.2}%
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\contentsline {section}{\numberline {5.7}Linear Combinations of ANNs and Their Properties}{93}{section.5.7}%
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\contentsline {section}{\numberline {5.8}Neural Network Diagrams}{103}{section.5.8}%
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\contentsline {chapter}{\numberline {6}ANN Product Approximations}{106}{chapter.6}%
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\contentsline {section}{\numberline {6.1}Approximation for Products of Two Real Numbers}{106}{section.6.1}%
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\contentsline {subsection}{\numberline {6.1.1}The squares of real numbers}{107}{subsection.6.1.1}%
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\contentsline {subsection}{\numberline {6.1.2}The $\prd $ network}{118}{subsection.6.1.2}%
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\contentsline {section}{\numberline {6.2}Higher Approximations}{123}{section.6.2}%
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\contentsline {subsection}{\numberline {6.2.1}The $\tun $ Neural Networks and Their Properties}{124}{subsection.6.2.1}%
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\contentsline {subsection}{\numberline {6.2.2}The $\pwr $ Neural Networks and Their Properties}{129}{subsection.6.2.2}%
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\contentsline {subsection}{\numberline {6.2.3}The $\tay $ Neural Networks and Their Properties}{139}{subsection.6.2.3}%
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\contentsline {subsection}{\numberline {6.2.4}Neural Network Approximations For $e^x$.}{144}{subsection.6.2.4}%
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\contentsline {chapter}{\numberline {7}A modified Multi-Level Picard and Associated Neural Network}{145}{chapter.7}%
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\contentsline {chapter}{\numberline {8}ANN first approximations}{148}{chapter.8}%
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\contentsline {section}{\numberline {8.1}Activation Function as Neural Networks}{148}{section.8.1}%
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\contentsline {section}{\numberline {8.2}ANN Representations for One-Dimensional Identity}{149}{section.8.2}%
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\contentsline {section}{\numberline {8.3}Modulus of Continuity}{158}{section.8.3}%
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\contentsline {section}{\numberline {8.4}Linear Interpolation of Real-Valued Functions}{158}{section.8.4}%
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\contentsline {subsection}{\numberline {8.4.1}The Linear Interpolation Operator}{159}{subsection.8.4.1}%
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\contentsline {subsection}{\numberline {8.4.2}Neural Networks to Approximate the $\lin $ Operator}{160}{subsection.8.4.2}%
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\contentsline {section}{\numberline {8.5}Neural Network Approximations of 1-dimensional Functions.}{164}{section.8.5}%
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\contentsline {section}{\numberline {8.6}$\trp ^h$ and Neural Network Approximations For the Trapezoidal Rule.}{167}{section.8.6}%
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\contentsline {section}{\numberline {8.7}Linear Interpolation for Multi-Dimensional Functions}{170}{section.8.7}%
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\contentsline {subsection}{\numberline {8.7.1}The $\nrm ^d_1$ and $\mxm ^d$ Networks}{170}{subsection.8.7.1}%
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\contentsline {subsection}{\numberline {8.7.2}The $\mxm ^d$ Neural Network and Maximum Convolutions }{176}{subsection.8.7.2}%
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\contentsline {subsection}{\numberline {8.7.3}Lipschitz Function Approximations}{180}{subsection.8.7.3}%
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\contentsline {subsection}{\numberline {8.7.4}Explicit ANN Approximations }{182}{subsection.8.7.4}%
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\contentsline {part}{III\hspace {1em}A deep-learning solution for $u$ and Brownian motions}{184}{part.3}%
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\contentsline {chapter}{\numberline {9}ANN representations of Brownian Motion Monte Carlo}{185}{chapter.9}%
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\contentsline {subsection}{\numberline {9.0.1}The $\mathsf {E}$ Neural Network}{188}{subsection.9.0.1}%
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\contentsline {subsection}{\numberline {9.0.2}The $\mathsf {UE}$ Neural Network}{193}{subsection.9.0.2}%
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\contentsline {subsection}{\numberline {9.0.3}The $\mathsf {UEX}$ network}{197}{subsection.9.0.3}%
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\contentsline {subsection}{\numberline {9.0.4}The $\mathsf {UES}$ network}{201}{subsection.9.0.4}%
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\contentsline {section}{\numberline {9.1}Bringing It All Together}{203}{section.9.1}%
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\contentsline {chapter}{\numberline {10}Conclusions and Further Research}{204}{chapter.10}%
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\contentsline {section}{\numberline {10.1}Further operations and further kinds of neural networks}{204}{section.10.1}%
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\contentsline {subsection}{\numberline {10.1.1}Mergers and Dropout}{204}{subsection.10.1.1}%
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\contentsline {section}{\numberline {10.2}Code Listings}{206}{section.10.2}%
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