clt_dissertation_draft_v10.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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\newacronym{unc}{UNC}{The University of North Carolina at Chapel Hill}
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%\frontmatter

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

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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}
\include{slide-NeuralNetworks.tex}
\include{NNFEM.tex}


\chapter{PINNs and DNN Solution Methods for PDE}
%\input{ch-PINN.tex}
%\include{ch-PINNModulus.tex}
%\include{slide-UseCasePINNs.tex}
%\include{ch-PINNs.tex}
%\include{ch-Methods.tex}
%\include{slide--PINNConvergence.tex}
%% \include{ref-ModulusPDEStandards.tex}
%\include{ch-NeuralTangentKernel.tex}
%\include{sec-PINNsAdvancedTechniques.tex}
%% \include{ref-PINNConvergence.tex}
%\include{PINNConvergence.tex}

\chapter{Least Squares Finite Element Methods}
%\input{~/git/IBPINN/unc-dissertation/lsfem/sec-LSFEM-all.tex}
%\include{ch-LeastSquaredFiniteElementMethod.tex}
%\include{sec-LSFEM-part0.tex}
%\include{ch-LSFEM.tex}
%\include{ch-FOSLSElliptic.tex}
%\include{sec-LSFEMDataAssimilation}


\chapter{Stokes Equations}
\include{ch-Stokes_FOSLS.tex}

%\chapter{FOSLS-PINNs}

\chapter{Elliptic or Poisson FOSLS Background}
\include{./notes/FOSLS/ch-DeepFOSLS1D-Formulations.tex}
\include{./notes/FOSLS/sec-EllipticFOSLS-Results.tex}
\include{./notes/FOSLS/ch-DeepFOSLS2D-Formulations.tex}


\chapter{Elliptic FOSLS Results}
\include{./notes/FOSLS/ch-DeepFOSLS1D2D-Results.tex}




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

\section{Solid Mechanics: Linear Elasticity, Pure Traction FOSLS Formulation}


\section{Solid Mechanics: Linear Elasticity Pure Displacement FOSLS Formulation}


\section{Solid Mechanics: Linear Elasticity General FOSLS Formulation}


\section{Solid Mechanics: Linear Elasticity Mixed Formulation}


\section{Solid Mechanics: Hyperelasticity Mixed Formulation}



\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}


\chapter{FOSLS Formulations: Solid Mechanics}
%\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{Miscellaneous Tables}
%\include{newmaterialmisc.tex}

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


\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}

\section{Channel Flow}
\subsection{Hagen-Poiseuille }
\include{./notes/FOSLS/ch-SteadyNavierStokes2D-FOSLS-Results.tex}
\include{./notes/FOSLS/sec-HagenPoiseuilleChannel2D.tex}
\subsection{Couette }
\include{./notes/FOSLS/sec-CouetteChannel2D.tex}

\section{Kovasznay}
\include{./notes/FOSLS/sec-Kovasznay2D.tex}

\include{./notes/FOSLS/sec-BercovierSquare2D.tex}

\section{Ruas Disk}
%\include{./notes/FOSLS/sec-RuasDisk2D.tex}

\section{2D Steady Newtonian Taylor-Green Vortex}
\include{./notes/FOSLS/sec-TaylorGreen2D.tex}


\section{Chip/Block Stenosis}
%\begin{frame}{Convergence: RELU, Adam, Block 2D, VPS}
\input{/home/chaztikov/git/modulus_examples/fosls_examples/chip_2d/chip_2d/results/chip_2d-8-5-adam-vps-table_L2_relative_errors_.tex}
%\end{frame}
\include{./notes/FOSLS/sec-BlockStenosisChannel2D.tex}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-Block2D.tex}




%
\include{./notes/FOSLS/ch-NavierStokes2D-FOSLS-Results.tex}

%
%\chapter{Numerical Experiments: Steady Navier-Stokes, 3D}
\include{./notes/FOSLS/ch-SteadyNavierStokes3D-FOSLS-Results.tex}
\include{./notes/FOSLS/sec-TaylorGreen3D.tex}
\include{./notes/FOSLS/sec-ABCFlow3D.tex}

%\chapter{Numerical Experiments: Unsteady Navier Stokes, 2D}
%\include{./notes/FOSLS/sec-Cylinder2DT.tex}
%\include{./notes/FOSLS/sec-TaylorGreen2DT.tex}

%\chapter{Numerical Experiments: Unsteady Navier-Stokes, 3D}
%\include{./notes/FOSLS/ch-NavierStokes3D-FOSLS-Results.tex}
%\include{./notes/FOSLS/sec-Beltrami3DT.tex}
%\include{./notes/FOSLS/sec-TaylorGreen3DT.tex}




\section{Numerical Experiments: Elliptic and Poisson}


\section{Numerical Experiments: Linear Elasticity}
\include{LEFormulationComparison.tex}
\include{LESoftHardBC.tex}
\subsection{Plane Strain Block 2D}
\include{slide-PlaneDisplacement.tex}
\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, 2D}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-LDC2D.tex}
\include{misc-pytables.tex}






%\include{./notes/FOSLS/sec-AnnularRing2D.tex}
%\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing.tex}
%\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing1.tex}

\chapter{Inverse Problems}
\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}


\section{Plane Stress}
\include{sec-LinearElasticity2D-Results-PlaneStress}
\include{sec-LinearElasticity2D-Results-PlaneStress-NTK}
\include{sec-LinearElasticity2D-Results-PlaneStress-ImportanceSampling}

We consider a rectangular domain in 2D governed by solid mechanics with a linear elastic constitutive model.  Plane stress constraints imposed on the 2D block of material. 



In primitive variables, the Navier equations describe the response of the material to pure Neumann (applied stress) boundary conditions and yield a set of second order linear PDEs to solve in terms of the displacement field with $d$ components, displacement).

An alternative in this case (constraints) to solving the Navier Equations is to use either Mixed or FOS formulations. 

Here, we compare the Velocity-Flux formulation with its 2-stage variant as described in \ref{sec:LinearElasticityVelocityGradientPlaneStress2Stage}

\begin{itemize}
	\item Velocity-Flux (1-stage)
	\item Velocity-Flux (2-stage)
	\item Stress-Displacement
\end{itemize}
formulations.  


\subsubsection{Formulation Comparison}
We compare the convergence of errors in the approximation obtained by each formulation with respect to increasing number of degrees of freedom, governed by the underlying width and depth (hence number of NN weights/parameters) of the DNN.


\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/
\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/multirun
\end{verbatim}
%\begin{verbatim}



\subsubsection{Using NTK}
We focus on the first phase of the 2-stage velocity-flux formulation, and compare preceding results with the use of the NTK to determine/assign values of (constant within each iteration) the scalar factors serving as relative weights of individual components of the discrete loss functional of the interior and (with Neumann conditions) boundary.
%\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/ntk/vvg1/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=6,custom.activation=5,custom.fosls_type=vvg1,network_dir=checkpoint_plane_stress_pinn_fosls_importance_vvg1/checkpoint_plane_stress_pinn_fosls_importance_vvg1/validators/
\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/ntk
\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/ntk/vvg1/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=6,custom.activation=5,custom.fosls_type=vvg1,network_dir=checkpoint_plane_stress_pinn_fosls_importance_vvg1/checkpoint_plane_stress_pinn_fosls_importance_vvg1/validators/
\end{verbatim}
%\begin{verbatim}



\subsubsection{Using Importance Sampling}
%\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/importance_sampling/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=6,custom.activation=5,custom.fosls_type=vvg1
\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/importance_sampling
\end{verbatim}
%\begin{verbatim}

\section{L-Shape Domain}
\include{slide-LShape-Intro.tex}

\section{AnnularRing}
\include{slide-AnnularRing.tex}
\include{./notes/FOSLS/sec-AnnularRing2D.tex}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing.tex}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing1.tex}
%\input{./notes/FOSLS/SECTION_ANNULARRING_STEADY_2D.tex}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing1.tex}
\include{./notes/FOSLS/sec-AnnularRing2D.tex}
\include{./notes/FOSLS/ch-DeepFOSLS-Results-AnnularRing.tex}
\include{./notes/FOSLS/SECTION_ANNULUARRING_PARAMETERIZED_STEADY_2D.tex}
\include{./notes/FOSLS/SECTION_ANNULUARRING_PRESSUREMONITORS_STEADY_2D.tex}
\include{./notes/FOSLS/SECTION_ANNULUARRING_STEADY_2D.tex}

%\chapter{PIGAN with FOSLS}
%\include{./notes/FOSLS/PIGANDeepFOSLS.tex}

\section{SteadyChannel2D}
\include{slide-SteadyChannel2D.tex}

\section{BercovierEngelman2D}
% \include{slide-BercovierEngelman2D-0.tex}
\include{slide-BercovierEngelman2D.tex}
\include{slide-BercovierEngelman2D-SILU.tex}
\include{slide-BercovierEngelman2D-NTK-SILU.tex}

\section{RuasDisk2D}
\include{slide-RuasDisk2D.tex}

\section{Cylinder2D}
\include{slide-Cylinder2D-Intro.tex}
\include{slide-Cylinder2D.tex}
\include{slide-Cylinder2D-RELU.tex}
\include{slide-Cylinder2D-SILU.tex}
%Cylinder Tables
%\include{/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/results/cylinder_2d-tables.tex}
\include{./physical_systems/cylinder_2d/results/cylinder_2d-tables.tex}
%-rw-rw-r-- 1 chaztikov chaztikov 17766 Dec 12 06:36 \include{./notes/FOSLS/ch-DeepFOSLS-Results-Cylinder2D.tex}

\include{./notes/FOSLS/ch-DeepFOSLS-Results-Cylinder2D.tex}
\include{./notes/FOSLS/sec-CylinderChannel2D.tex}
%\include{./notes/FOSLS/SEC-CYLINDER-STEADY-2D}
\include{./notes/FOSLS/SECTION_CYLINDER_STEADY_2D.tex}
\include{./notes/FOSLS/SECTION_SDF_CYLINDER_STEADY_2D.tex}
\include{./notes/FOSLS/SEC_CYLINDER_STEADY_2D.tex}
\include{./notes/FOSLS/SEC-CYLINDER-STEADY-2D.tex}
\include{./notes/FOSLS/viscoelastic_oldroydb_cylinder_2D_2D.tex}

%\end{verbatim}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=4,custom.fosls_type=vp,network_dir=network_checkpoint_cylinder_2d_fosls_vp/network_checkpoint_cylinder_2d_fosls_vp/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=1,custom.fosls_type=vps,network_dir=network_checkpoint_cylinder_2d_fosls_vps/network_checkpoint_cylinder_2d_fosls_vps/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=2,custom.fosls_type=vps,network_dir=network_checkpoint_cylinder_2d_fosls_vps/network_checkpoint_cylinder_2d_fosls_vps/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=3,custom.fosls_type=vps,network_dir=network_checkpoint_cylinder_2d_fosls_vps/network_checkpoint_cylinder_2d_fosls_vps/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=4,custom.fosls_type=vps,network_dir=network_checkpoint_cylinder_2d_fosls_vps/network_checkpoint_cylinder_2d_fosls_vps/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=4,custom.fosls_type=vvg,network_dir=network_checkpoint_cylinder_2d_fosls_vvg/network_checkpoint_cylinder_2d_fosls_vvg/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=4,custom.fosls_type=vvp,network_dir=network_checkpoint_cylinder_2d_fosls_vvp/network_checkpoint_cylinder_2d_fosls_vvp/validators/
/home/chaztikov/git/modulus_examples/fosls_examples/cylinder_2d/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=4,custom.fosls_type=vvt,network_dir=network_checkpoint_cylinder_2d_fosls_vvt/network_checkpoint_cylinder_2d_fosls_vvt/validators/
\end{verbatim}
%\begin{verbatim}


\section{LDC2D}
\include{slide-LDC2D.tex}
\include{slide-LDC2D-SILU.tex}
\include{slide-LDC2D-RELU.tex}
\include{slide-LDC2D-TANH.tex}

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



\section{TaylorGreenVortex2D}
\begin{verbatim}
df.to_latex(buf=tablewritename,caption=Path(tablewritename).stem.replace('_',' ').replace('-',' '),label=Path(tablewritename).stem)	
/home/chaztikov/git/modulus_examples/fosls_examples/taylorgreenvortex_2d/mu=0.01
/home/chaztikov/git/modulus_examples/fosls_examples/taylorgreenvortex_2d/mu=0.01/multirun/arch.fully_connected.layer_size=512,arch.fully_connected.nr_layers=6,custom.activation=10,custom.fosls_type=vp,network_dir=tgvortex_fosls_vp/
/home/chaztikov/git/modulus_examples/fosls_examples/taylorgreenvortex_2d/mu=0.01/multirun/arch.fully_connected.layer_size=128,arch.fully_connected.nr_layers=6,custom.activation=5,custom.fosls_type=vvp,network_dir=tgvortex_fosls_vvp/
\end{verbatim}








\include{slide-SteadyNSFormulations.tex}
\include{slide-SteadyNSPressureRecovery.tex}
% \include{slide-test.tex}

\include{slide-WeightedLSFEM.tex}
\begin{verbatim}
/home/chaztikov/git/modulus_examples/fosls_examples/regularizedtanh_ldc_2d/sdf
/home/chaztikov/git/modulus_examples/fosls_examples/regularizedtanh_ldc_2d/foslsfunctional
/home/chaztikov/git/modulus_examples/fosls_examples/SolidMechanics/LinearElasticity/plane_stress/ntk/vvg1
\end{verbatim}

\chapter{Conclusions and Discussion}
\include{slide-Discussion.tex}
\include{slide-Discussion-PINNOverfitting.tex}
3.3. Approximation Properties of ReLU-NNs: h-Galerkin BEM 11
DNN Architecture. We use a feed-forward DNN[1] with 12 fully-connected layers. After 8 layers, the network splits into 6 branches, 4 layers each: the first branch decides about the optimal nref parameter - $h$ refinement, the remaining branches decide about modifying the polynomial orders $-p$ refinement. Experiments have shown that further expanding of the network makes it prone to overfitting [8]. Splitting the network into branches assures sufficient parameter freedom for each variable. This approach also simplifies the model: there is no need to train a DNN for each variable. Since all possible decisions are encoded as categorical variables, we use cross-entropy as the loss function. We encoded the

 Salman, S., Liu, X.: Overfitting mechanism and avoidance in deep neural networks.
arXiv preprint arXiv:1901.06566 (2019)
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% Overview of Neural Networks and Supervised (Learning) NN Approximation
%\chapter{Chapter 1}
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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}
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% Brief Introduction to equation-informed (unsupervised) NN approximations to PDEs PINNs, overview of results (forward, backward, identification, SDEs), FFR Application?
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%\input{chapter2.tex}
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% 1D Deep LS, Deep FOSLS, Deep Energy Method comparison
%\chapter{Chapter 2}
%\input{chapter2.tex}
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% 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}
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% 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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% 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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% 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}
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% PINN Approximations to Navier-Stokes: 
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%\chapter{PINN Approximations to Navier-Stokes: Steady Newtonian Fluid Mechanics}
%\input{chapter2.tex}
%\subimport{notes/}{windkessel_overview.tex}
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% Time-Dependent FOSLS Method, 1D/2D Advection-Diffusion-Reaction, Burger's, KdV, NL Schrodinger
%\chapter{Chapter 2}
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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
%\chapter{Chapter 2}
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% 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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%% BIBLIOGRAPHY AND OTHER LISTS
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%\backmatter

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

% Add bibliography to contents page
%\addcontentsline{toc}{chapter}{Bibliography} %'Bibliography' into toc

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


\end{document}