clt_oral_presentation_v2.tex

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\title{Dissertation Title}
\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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\newacronym{unc}{UNC}{The University of North Carolina at Chapel Hill}
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\linespread{1.0}
\begin{document}
	


	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%
	%
	%
	%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	LSFEM Review Fluid Mechanics
%	Navier-Stokes (Incompressible Newtonian Fluids)
%	VVP
%	VVG
%	VVT
%	(Variable Viscosity e.g. Carreau) Navier-Stokes (Incompressible Newtonian Fluids)
%	VPS
%	PINN: Linear Elasticity Plane Displacement
%	Linear elasticity: Mixed Formulation
%	(Spring 2022)
%	PINN Application: Navier-Stokes Idealized Vessels
%	Newtonian
%	Carreau
%	Singly and Doubly Stenotic
%	FOSLS-PINN: Linear Elasticity Plane Stress
%	Linear elasticity: Mixed Formulation
%	Linear elasticity: VVG Formulation
%	PINN and FOSLS-PINN: Channel Flow 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Sampled Points
%	PINN and FOSLS-PINN: Bercovier-Engelman 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Steady Cylinder 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Block Stenosis 2D
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Lid Driven Cavity 2D
%	Importance Sampling
%	Signed Distance Function Weighting (Corner Singularities)
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	PINN and FOSLS-PINN: Annular Ring 2D
%	Regular
%	Hard BCs
%	Residual Gradient
%	VP
%	VPS
%	VVG
%	VVT
%	VVP
%	Convergence w.r.t. Number of Neurons (Depth)
%	Convergence w.r.t. Number of Neurons (Width)
%	Convergence w.r.t. Number of Neurons (Size)
%	Variational PINN (VPINN)
%	Poisson, Laplace, or Linear Elasticity
%	VPINN
%	FOSLS VPINN
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 0
%	% Everything, disorganized	
%%	\begin{frame}{Chapter 0}
%%	\end{frame}
%	\begin{frame}{
%%			\include{}
%		}
%	\end{frame}
%\include{slide-UseCasePINNs.tex}

\begin{frame}{Universal Approximation Theorem}
	
\end{frame}





\begin{frame}{Physics-Informed Neural Networks}
	
\end{frame}


\begin{frame}{PINN Applications, Use Cases}
	%	\include{slide-UseCasePINNs.tex}
	%\section{Use Cases of PINNs vs Classical Numerical Methods}
	Current AI methods will be slower than the traditional solvers for training of a single case/geometry. 
	\begin{enumerate}
		\item Multiple (parameterized) cases where there are several different configurations in the analysis space
		\item Parametric Models
		%different model configurations (subject to physical constraints) based on a multitude of parameters.
		%Compare autlod: Appearance-Driven Automatic 3D Model Simplification
		\item Surrogate Models 
		\item Design Optimization
		\item Inverse Problems
		\item Reduced-Order Models
		%		(data is given but determine coefficients of PDEs)
		\item Data Assimilation (Sparse/Incomplete Field Measurements, Digital Twins, Medical Imaging, Full Waveform Inversion)
		%Cases where measured data is available but the Physics is too complicated or not fully understood
		%\item Data assimilation cases where there is measured or simulated data for some variables but not the entire field (e.g. digital twins, medical imaging, full waveform inversion etc.)
		\item Surrogate for compute intensive model components (constitutive models, turbulence models)
		%part of the solver and call that from the solver during the analysis to improve the speed (constitutive models, turbulence models or radiation viewfactors etc.)
		\item Transfer Learning
		%		\item Runge Kutta simulations (extreme order)
		%\item Point clouds can sometimes present higher quality solutions than mesh based simulations (since without convergence studies, mesh based solutions can yield questionable results).
	\end{enumerate}
\end{frame}


\begin{frame}{PINN Error Estimates and Solutions}
	
	Elliptic PDE
	\begin{enumerate}
		\item Error Estimates and Convergence of PINNs in Primitive Variables ()
		\item Error Estimates and Convergence of PINNs using LS and FOSLS functionals for 1D Elliptic Problems ()
	\end{enumerate}

	Stokes Equations
	\begin{enumerate}
		\item VPVNet: PINN using VPV formulation, convergence, and numerical examples
	\end{enumerate}

	Navier-Stokes Equations (Primitive Variables)
	\begin{enumerate}	
		\item Rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with physics informed neural networks (De Ryck et al)
		\item Underlying PDE residual can be made arbitrarily small for $\tanh$ neural networks with 2 hidden layers. (De Ryck et al)
		\item Total error can be estimated in terms of the training error, network size and number of quadrature points. (De Ryck et al) 
		\item The theory is illustrated with numerical experiments.
	\end{enumerate}

\end{frame}

\include{slide-NeuralNetworks.tex}

\include{slide-NeuralNetworksFEM.tex}

\include{slide-LSFEM.tex}

\include{slide-WeightedLSFEM.tex}
%
%\begin{frame}{Weighted LSFEM for Singular PDEs}
%	FEM effectiveness limited by locally non-smooth solutions 
%	%lack smoothness on a relatively small subset of the domain. 
%	Standard $L^2$ LSFEM: for problems with singular solutions, suboptimal rate of convergence (fail)
%	Weighted LSFEM: enhance norm used in the LS functional with weight functions, recover near-optimal convergence 
%	
%	PDE with singular behavior at isolated locations the domain. 
%	Problems with smooth data may fail to provide smooth solutions (b.c. domain or operator). 
%	Example
%	\begin{align*}
%		\left\{\begin{aligned}
%			\mathcal{K}(u) & =f & \text { in } \Omega \\
%			u & =g & \text { on } \partial \Omega,
%		\end{aligned}\right.
%	\end{align*}
%	where $\mathcal{K}$ is a second-order differential operator. If $f \in L^2(\Omega)$ and $g \in H^{3 / 2}(\Omega)$ is sufficient to guarantee that $u \in H^2(\Omega)$, then we consider the problem to have full regularity. We consider problems without this property to have a low regularity, or (potentially) nonsmooth solutions. For example, Poisson's equation is known to have full regularity when $\Omega$ is convex, but can have nonsmooth solutions when $\partial \Omega$ has corners (or edges) with interior angle greater than $\pi$ [19]. This lack of smoothness is localized, however. In any subdomain excluding a neighborhood of each corner point the solution remains smooth. Other elliptic problems have similar behavior as a consequence of the domain, see e.g., $[24,25]$. 
%	
%	The operator $\mathcal{K}$ can also induce a loss of smoothness when coefficients are either singular (i.e., $\rightarrow \infty$ ) or degenerate (i.e., $\rightarrow 0$ ) at distinct points in $\Omega[5]$.
%	
%	Invariably, numerical methods tend to suffer as a consequence of a loss of regularity. Finite element convergence rates can be reduced or, in some cases, the method can fail to converge to the solution of the problem. 
%	Often: localized non-smooth solution behavior yield Sub-optimal global convergence rates: \textbf{pollution effect}
%\end{frame}


\include{slide-SteadyNSFormulations.tex}


%\begin{frame}{Overview: PINN and Use Cases}
%\end{frame}
%
%\begin{frame}{PINNs Intro}
%\end{frame}
%
%\begin{frame}{PINNs Review of Applications}
%\end{frame}
%
%\begin{frame}{NN and FEM Review}
%\end{frame}
%
%\begin{frame}{PINN: Convergence Review}
%\end{frame}
\include{slide--PINNConvergence}

%	\begin{frame}{FOSLS-PINN Motivation}
	%	\end{frame}
%	\begin{frame}{LSFEM Review Elasticity}
	%	\end{frame}

%\begin{frame}{Stokes/Linear Elasticity}
%\end{frame}
%
%\begin{frame}{Pure Traction}
%\end{frame}
%
%\begin{frame}{Pure Displacement}
%\end{frame}
%
%\begin{frame}{Mixed Formulations for Linear Elasticity}
%\end{frame}
%
%\begin{frame}{Mixed Formulations for Hyperelasticity}
%	
%\end{frame}

\include{slide-LinearElasticity}

\include{slide-SteadyChannel2D.tex}
\include{slide-BercovierEngelman2D-Intro.tex}
\include{slide-BercovierEngelman2D.tex}
\include{slide-BercovierEngelman2D-SILU.tex}
\include{slide-BercovierEngelman2D-NTK-SILU.tex}
\include{slide-BercovierEngelman2D-NTK-RELU.tex}

\include{slide-PlaneDisplacement-Intro}
\include{slide-PlaneDisplacement}

\include{slide-Chip2D-Intro.tex}
\include{slide-Chip2D}

\include{slide-Cylinder2D-Intro.tex}
%\include{slide-Cylinder2D}
\include{slide-Cylinder2D-SILU}
\include{slide-Cylinder2D-RELU}

\include{slide-LDC2D}
\include{slide-LDC2D-SILU}
\include{slide-LDC2D-RELU}
%\include{slide-LDC2D-TANH}

\include{slide-AnnularRing}
\include{appendix-misc.tex}

%	\begin{frame}{
%			\include{slide-PINNs.tex}
% TODO: \usepackage{graphicx} required

%		}
%	\end{frame}
%	\begin{frame}{
%			\include{slide-Methods.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{slide-LSFEM.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS1D-Formulations.tex}
%	\end{frame}
%	%\begin{frame}{
%			\include{./notes/FOSLS/sec-EllipticFOSLS-Results.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS2D-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-LinearElasticityFOSLS-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-LinearElasticityFOSLS-Results.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-HyperElasticityFOSLS-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-HyperElasticityFOSLS-Results.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-VVP-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-VVG-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-VVT-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-Formulations.tex}
%	%\begin{frame}{
%			\include{./notes/FOSLS/slide-StokesFOSLS-Results.tex}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-LinearElasticityStokesFOSLS-Formulations.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureDisplacement.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-PureTraction.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-LinearElasticityStokesFOSLS-Formulations-Mixed.tex}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-HyperElasticityFOSLS-Formulations.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-HyperElasticityFOSLS-Formulations-Mixed.tex}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-NavierStokesFOSLS-Formulations.tex}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-NavierStokesFOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-SteadyNavierStokes2D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-HagenPoiseuilleChannel2D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-CouetteChannel2D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-KovasznayRectangle2D.tex}
%		}
%	\end{frame}
%	
%%	\begin{frame}{
%			\include{./notes/FOSLS/sec-BercovierSquare2D.tex}
%%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-RuasDisk2D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen2D.tex}
%		}
%	\end{frame}
%	
%%	\begin{frame}{
%			\include{./notes/FOSLS/sec-CylinderChannel2D.tex}
%%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-BlockStenosisChannel2D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-AnnularRing2D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-SteadyNavierStokes3D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen3D.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-ABCFlow3D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-NavierStokes2D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen2DT.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-NavierStokes3D-FOSLS-Results.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-Beltrami3DT.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-TaylorGreen3DT.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/sec-Cylinder2DT.tex}
%		}
%	\end{frame}
%	
%	
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-LDC2D.tex}
%		}
%	\end{frame}
%
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Cylinder2D.tex}
%		}
%	\end{frame}
%
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Block2D.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-AnnularRing.tex}
%		}
%	\end{frame}
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-AnnularRing1.tex}
%		}
%	\end{frame}
%	
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Cylinder2D_inverse.tex}
%		}
%	\end{frame}
%	
%	\begin{frame}{
%			\include{./notes/FOSLS/slide-DeepFOSLS-Results-Block2D_inverse.tex}
%		}
%	\end{frame}
%
%%	\input{./slide-Chapter0.tex}
%	%\input{./notes/FOSLS/SECTION_ANNULUARRING_PRESSUREMONITORS_STEADY_2D}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 1
%	% Overview of Neural Networks and Supervised (Learning) NN Approximation
%	\begin{frame}{Chapter 1}
%		}
%	\end{frame}
%
%	%\input{chapter1.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% 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, 
%	\begin{frame}{Chapter 2}
%		}
%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 3
%	% Brief Introduction to equation-informed (unsupervised) NN approximations to PDEs PINNs, overview of results (forward, backward, identification, SDEs), FFR Application?
%	\begin{frame}{Chapter 3}
%		}
%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
%	\begin{frame}{Chapter 2}
%		}
%	\end{frame}
%
%	%\input{chapter2.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 2D (Unconstrained?) PDEs
%	% Deep FOSLS vs PINNs comparison.  
%	% Poisson-like Equations
%	% Poisson
%	% Diffusion
%	% Helmholtz
%	%
%	\begin{frame}{Chapter 2}
%		}
%	\end{frame}
%
%	%laplace
%	%\input{poisson2d.tex}
%	%\input{surfacepoisson2d.tex}
%	%%\input{diffusion2d.tex}
%	%%\input{diffusionreaction2d.tex}
%	%singularlyperturbed
%	%\input{advectiondiffusionreaction2d.tex}
%	%\input{helmholtz2d.tex}
%	%\input{chapter2.tex}
%	
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% 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
%	\begin{frame}{Finite Element Methods and Least-Squares FEM in Solid and Fluid Mechanics}
%		}
%	\end{frame}
%
%	%%\input{Galerkin and mixed Galerkin finite elements.tex}
%	%\input{GalerkinandMixedGalerkinFiniteElements.tex}
%	%%\input{Least-squares mixed finite elements.tex}
%	%\input{LeastSquaresMixedFiniteElements.tex}
%	%%\input{Outline}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% 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
%	\begin{frame}{Continuum mechanical background}
%		}
%	\end{frame}
%	%\input{KinematicsAndDeformationMeasures.tex}
%	%\input{StressQuantities.tex}
%	%\input{Hyperelasticmaterials.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Background: LSFFEM for Steady Newtonian Fluid Mechanics
%	\begin{frame}{First-Order Systems Least-Squares Formulations of Navier-Stokes Equations for Steady Newtonian Fluid Mechanics}
%		}
%	\end{frame}
%	%\input{LinearElasticityPureTractionFormulation.tex}
%	%\input{LinearElasticityPureDisplacementFormulation.tex}
%	%\input{LinearElasticityTractionDisplacementFormulation.tex}
%	%\input{hyperelasticfosls.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% Background: LSFFEM for Steady Newtonian Fluid Mechanics
%	\begin{frame}{First-Order Systems Least-Squares Formulations of Navier-Stokes Equations for Steady Newtonian Fluid Mechanics}
%		}
%	\end{frame}
%	%\input{VPFormulation.tex}
%	%\input{VVFormulation.tex}
%	%\input{VVPFormulation.tex}
%	%\input{VPSFormulation.tex}
%	%\input{VVGFormulation.tex}
%	%\input{VVTFormulation.tex}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% PINN Approximations to Navier-Stokes: 
%	% 2D Steady Newtonian Fluid Mechanics
%	\begin{frame}{PINN Approximations to Navier-Stokes: Steady Newtonian Fluid Mechanics}
%		}
%	\end{frame}
%	%\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}
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	%
%	%
%	%
%	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	% Chapter 
%	% 2D Steady Non-Newtonian Fluid Mechanics, Special Methods, Applications
%%	\begin{frame}{Chapter 2}
%%		}
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%	% Time-Dependent FOSLS Method, 1D/2D Advection-Diffusion-Reaction, Burger's, KdV, NL Schrodinger
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%	% PIGAN FOSLS vs Primitive FOSLS: Pressure Reconstruction, Darcy, and Elasticity
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%	% Bayesian PINNs vs Bayesian FOSLS PINNs for UQ, noise distributions
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%	% Operator Learning (DeepONet, MIONet, and Fourcastnet) 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
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%	%% BIBLIOGRAPHY AND OTHER LISTS
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%	% Bibliography
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%	%% APPENDICES
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