clt_dissertation_draft_v3.tex

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\title{Deep Neural Network Solution Methods for First Order Least Squares Formulations of PDEs in Fluid and Solid Mechanics}
\author{Dissertation Author}
%\committee{Advisor}{Committee Member}{Committee Member}{Committee Member}{Committee Member}
\date{January 1, 1970}

%\input{abstract.tex}
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%\dedication{To Wawa}

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% - pdflatex dissertation
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\newacronym{unc}{UNC}{The University of North Carolina at Chapel Hill}
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%% INITIAL THINGS
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%\frontmatter

%\maketitle

\chapter*{Acknowledgements}
\input{acknowledgements.tex}

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%\listofabbreviations

%\mainmatter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%MAIN CHAPTERS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\pagestyle{plain} % Restore page numbers
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%
%
%
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% Chapter 0
% Everything, disorganized
%\chapter{Chapter 0}
%\include{UseCasePINNs}
\chapter{Introduction}
\include{ch-Introduction.tex}

\chapter{Neural Networks: Architectures, Activations, and Optimization}
\include{ch-NeuralNetworks.tex}
\include{NeuralNetworks.tex}
\include{sec-DeepReLUConvergence.tex}

\chapter{PINNs and DNN Solution Methods for PDE}
\include{ch-PINNModulus.tex}
\include{ch-PINNs.tex}
\include{ch-Methods.tex}
% \include{ref-ModulusPDEStandards.tex}
\include{ch-NeuralTangentKernel.tex}
\include{sec-PINNsAdvancedTechniques.tex}
\chapter{Least Squares Finite Element Methods}
\include{ch-LeastSquaredFiniteElementMethod.tex}
\include{ch-LSFEM.tex}
\include{ch-FOSLSElliptic.tex}
\include{sec-LSFEMDataAssimilation}

\chapter{FOSLS-PINNs}


\chapter{Fluid Mechanics Background}
\include{notes/FOSLS/ch-NavierStokesFormulations}
% \include{notes/FOSLS/NavierStokesFormulations}
%\include{notes/FOSLS/ch-NavierStokesFOSLS}

\chapter{Solid (Continuum) Mechanics Background}
\include{./notes/FOSLS/sec-LinearElasticity.tex}
\include{ch-ContinuumMechanicsHyperelastic.tex}


\chapter{Solid (Continuum) Mechanics Background}
\include{./notes/FOSLS/ch-DeepFOSLS1D-Formulations.tex}
\include{./notes/FOSLS/sec-EllipticFOSLS-Results.tex}
\include{./notes/FOSLS/ch-DeepFOSLS2D-Formulations.tex}

\chapter{Linear Elasticity Results}
\include{./notes/FOSLS/ch-LinearElasticityFOSLS-Formulations.tex}
\include{./notes/FOSLS/ch-LinearElasticityFOSLS-Results.tex}
\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureTraction.tex}

\chapter{Hyperelasticity Results}
%\include{./notes/FOSLS/ch-HyperElasticityFOSLS-Formulations.tex}
%\include{./notes/FOSLS/ch-HyperElasticityFOSLS-Results.tex}
%\include{./notes/FOSLS/ch-StokesFOSLS-VVP-Formulations.tex}
%\include{./notes/FOSLS/ch-StokesFOSLS-VVG-Formulations.tex}
%\include{./notes/FOSLS/ch-StokesFOSLS-VVT-Formulations.tex}
%\include{./notes/FOSLS/ch-StokesFOSLS-Formulations.tex}
%\include{./notes/FOSLS/ch-StokesFOSLS-Results.tex}

\include{./notes/FOSLS/ch-LinearElasticityStokesFOSLS-Formulations.tex}
\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureDisplacement.tex}
\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureTraction.tex}
\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-Mixed.tex}

%\include{./notes/FOSLS/ch-HyperElasticityFOSLS-Formulations.tex}
%\include{./notes/FOSLS/sec-HyperElasticityFOSLS-Formulations-Mixed.tex}
\chapter{Navier-Stokes FOSLS Formulations}
\include{./notes/FOSLS/ch-NavierStokesFOSLS-Formulations.tex}
\chapter{Navier-Stokes FOSLS Results}
%\include{./notes/FOSLS/ch-NavierStokesFOSLS-Results.tex}
%\include{./notes/FOSLS/ch-NavierStokesFOSLS-Results}
\chapter{Elliptic FOSLS Results}
\include{./notes/FOSLS/ch-DeepFOSLS1D2D-Results.tex}
\include{./notes/FOSLS/ch-SteadyNavierStokes2D-FOSLS-Results.tex}
\include{./notes/FOSLS/sec-HagenPoiseuilleChannel2D.tex}
\include{./notes/FOSLS/sec-CouetteChannel2D.tex}
\include{./notes/FOSLS/sec-Kovasznay2D.tex}
%\include{./notes/FOSLS/sec-BercovierSquare2D.tex}
\include{./notes/FOSLS/sec-RuasDisk2D.tex}
\include{./notes/FOSLS/sec-TaylorGreen2D.tex}

\include{./notes/FOSLS/sec-CylinderChannel2D.tex}
\include{./notes/FOSLS/SEC-CYLINDER-STEADY-2D}
\include{./notes/FOSLS/sec-BlockStenosisChannel2D.tex}
\include{./notes/FOSLS/sec-AnnularRing2D.tex}

\include{./notes/FOSLS/ch-SteadyNavierStokes3D-FOSLS-Results.tex}
\include{./notes/FOSLS/sec-TaylorGreen3D.tex}
\include{./notes/FOSLS/sec-ABCFlow3D.tex}
%
\include{./notes/FOSLS/ch-NavierStokes2D-FOSLS-Results.tex}
\include{./notes/FOSLS/sec-TaylorGreen2DT.tex}
%
%\include{./notes/FOSLS/ch-NavierStokes3D-FOSLS-Results.tex}
%\include{./notes/FOSLS/sec-Beltrami3DT.tex}
%\include{./notes/FOSLS/sec-TaylorGreen3DT.tex}
\include{./notes/FOSLS/sec-Cylinder2DT.tex}


\include{NNFEM.tex}
\include{PINNConvergence.tex}
\include{LEFormulationComparison.tex}
\include{LESoftHardBC.tex}


\section{Numerical Experiments: Elliptic and Poisson}


\section{Numerical Experiments: Linear Elasticity}
\subsection{Plane Strain Block 2D}
\subsection{Plane Stress Block 2D}
\include{sec-LinearElasticity2D-Results-PureTraction}


\section{Numerical Experiments: Hyperelastic Solid Mechanics}
\subsection{Compressible Neohookean Block}
\subsection{Incompressible Mooney-Rivlin Block}



\section{Numerical Experiments: Steady Navier Stokes, }
\include{./notes/FOSLS/ch-DeepFOSLS-Results-LDC2D.tex}

\include{./notes/FOSLS/ch-DeepFOSLS-Results-Cylinder2D.tex}

\include{./notes/FOSLS/ch-DeepFOSLS-Results-Block2D.tex}

\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing.tex}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing1.tex}


\include{./notes/FOSLS/ch-DeepFOSLS-Results-Cylinder2D_inverse.tex}

\include{./notes/FOSLS/ch-DeepFOSLS-Results-Block2D_inverse.tex}

\input{./ch-Chapter0.tex}
%\input{./notes/FOSLS/SECTION_ANNULUARRING_PRESSUREMONITORS_STEADY_2D}

\include{slide-Activations.tex}
\include{slide-AnnularRing.tex}
% \include{slide-BercovierEngelman2D-0.tex}
\include{slide-BercovierEngelman2D-NTK-SILU.tex}
\include{slide-BercovierEngelman2D-SILU.tex}
\include{slide-BercovierEngelman2D.tex}
\include{slide-Chip2D-Intro.tex}
\include{slide-Chip2D.tex}
\include{slide-Cylinder2D-Intro.tex}
\include{slide-Cylinder2D.tex}
\include{slide-Cylinder2D-RELU.tex}
\include{slide-Cylinder2D-SILU.tex}
\include{slide-LDC2D.tex}
\include{slide-LDC2D-SILU.tex}
\include{slide-LDC2D-RELU.tex}
\include{slide-LDC2D-TANH.tex}
\include{slide-NeuralNetworks.tex}
\include{slide--PINNConvergence.tex}
% \include{ref-PINNConvergence.tex}
\include{slide-PlaneDisplacement.tex}
\include{slide-RuasDisk2D.tex}
\include{slide-SteadyChannel2D.tex}
\include{slide-SteadyNSFormulations.tex}
\include{slide-SteadyNSPressureRecovery.tex}
% \include{slide-test.tex}
\include{slide-UseCasePINNs.tex}
\include{slide-WeightedLSFEM.tex}



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% Chapter 1
% Overview of Neural Networks and Supervised (Learning) NN Approximation
%\chapter{Chapter 1}
%\input{chapter1.tex}
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%
%
%
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% Chapter 2
% Brief Introduction to equation-informed (unsupervised) NN approximations to PDEs PINNs, overview of results (forward, backward, identification, SDEs), FFR Application? ROM and surrogate models, 
%\chapter{Chapter 2}
%\input{chapter2.tex}
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%
%
%
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% Chapter 3
% Brief Introduction to equation-informed (unsupervised) NN approximations to PDEs PINNs, overview of results (forward, backward, identification, SDEs), FFR Application?
%\chapter{Chapter 3}
%\input{chapter2.tex}
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%
%
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% Chapter 
% 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
%\chapter{Chapter 2}
%\input{chapter2.tex}
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% Chapter 
% 2D (Unconstrained?) PDEs
% Deep FOSLS vs PINNs comparison.  
% Poisson-like Equations
% Poisson
% Diffusion
% Helmholtz
%
%\chapter{Chapter 2}
%laplace
%\input{poisson2d.tex}
%\input{surfacepoisson2d.tex}
%%\input{diffusion2d.tex}
%%\input{diffusionreaction2d.tex}
%singularlyperturbed
%\input{advectiondiffusionreaction2d.tex}
%\input{helmholtz2d.tex}
%\input{chapter2.tex}

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% Chapter 
% Introduction: FEM and LSFEM methods
% Linear and Hyperelastic Elasticity
% PINNs vs Deep FOSLS (Deep Energy Method?)comparison
% thesisLeast-squares mixed finite elements for geometricallySteegerDiss.pdf
%\chapter{Finite Element Methods and Least-Squares FEM in Solid and Fluid Mechanics}
%%\input{Galerkin and mixed Galerkin finite elements.tex}
%\input{GalerkinandMixedGalerkinFiniteElements.tex}
%%\input{Least-squares mixed finite elements.tex}
%\input{LeastSquaresMixedFiniteElements.tex}
%%\input{Outline}
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% Chapter 
% 2D Static Solid Mechanics 
% Linear and Hyperelastic Elasticity
% PINNs vs Deep FOSLS (Deep Energy Method?)comparison
% thesisLeast-squares mixed finite elements for geometricallySteegerDiss.pdf
%\chapter{Continuum mechanical background}
%\input{KinematicsAndDeformationMeasures.tex}
%\input{StressQuantities.tex}
%\input{Hyperelasticmaterials.tex}
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%
%
%
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% Chapter 
% Background: LSFFEM for Steady Newtonian Fluid Mechanics
%\chapter{First-Order Systems Least-Squares Formulations of Navier-Stokes Equations for Steady Newtonian Fluid Mechanics}
%\input{LinearElasticityPureTractionFormulation.tex}
%\input{LinearElasticityPureDisplacementFormulation.tex}
%\input{LinearElasticityTractionDisplacementFormulation.tex}
%\input{hyperelasticfosls.tex}
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%
%
%
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% Chapter 
% Background: LSFFEM for Steady Newtonian Fluid Mechanics
%\chapter{First-Order Systems Least-Squares Formulations of Navier-Stokes Equations for Steady Newtonian Fluid Mechanics}
%\input{VPFormulation.tex}
%\input{VVFormulation.tex}
%\input{VVPFormulation.tex}
%\input{VPSFormulation.tex}
%\input{VVGFormulation.tex}
%\input{VVTFormulation.tex}
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%
%
%
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% Chapter 
% PINN Approximations to Navier-Stokes: 
% 2D Steady Newtonian Fluid Mechanics
%\chapter{PINN Approximations to Navier-Stokes: Steady Newtonian Fluid Mechanics}
%\input{chapter2.tex}
%\subimport{notes/}{windkessel_overview.tex}
%%\input{./notes/windkessel_overview.tex}
%%\input{../notes/FOSLS/BLOCKSTENOSISSTEADY2D.tex}
%%\input{/notes/FOSLS/BLOCKSTENOSISSTEADY2D.tex}
%%\input{../notes/FOSLS/BLOCKSTENOSIS_STEADY_2D.tex}
%%\input{../notes/FOSLS/SECTION_ANNULARRING_STEADY_2D.tex}
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%
%
%
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% Chapter 
% 2D Steady Non-Newtonian Fluid Mechanics, Special Methods, Applications
%\chapter{Chapter 2}
%\input{chapter2.tex}
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%
%
%
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% Chapter 
% Time-Dependent FOSLS Method, 1D/2D Advection-Diffusion-Reaction, Burger's, KdV, NL Schrodinger
%\chapter{Chapter 2}
%\input{chapter2.tex}
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%
%
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% Chapter 
% PIGAN FOSLS vs Primitive FOSLS: Pressure Reconstruction, Darcy, and Elasticity
%\chapter{Chapter 2}
%\input{chapter2.tex}
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%
%
%
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% Chapter 
% Bayesian PINNs vs Bayesian FOSLS PINNs for UQ, noise distributions
%\chapter{Chapter 2}
%\input{chapter2.tex}
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%
%
%
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% Chapter 
% Operator Learning (DeepONet, MIONet, and Fourcastnet) 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
%\chapter{Chapter 2}
%\input{chapter2.tex}
{\tiny }

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%\backmatter

% Bibliography
\bibliography{dissertation}
\addtocontents{toc}{\protect\vspace*{\baselineskip}}

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%\addcontentsline{toc}{chapter}{Bibliography} %'Bibliography' into toc

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\appendix
%\chapter{}
%\section{}


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