在 BEAMER 中出现以下错误

Question

您没有从日志中显示错误格式。

忽略丢失的图像，第一个错误是

! Undefined control sequence.
l.318 ... \lbrace x^k + \delta^k d : d \in \mathds
                                                  {D}^k_\Delta \rbrace $ is ...

?

添加

\usepackage{dsfont}
\newcommand\mathhd{\mathds}%??

然后你得到

! Double superscript.
l.322 ...^k$ and $\delta^{k+1} \gets \tau^ \delta^
                                                  k$.
?

因为你不能拥有x^2^3 我猜你的意思是\tau^{k}？

然后

! LaTeX Error: Unicode character   (U+205F)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.380 \end{frame}
                 
?

U+205F 是 Net 前面的空格，0 ) Net用空格代替

生产

! Missing \right. inserted.
<inserted text> 
                \right .
l.429 \end{frame}
                 
?

因为你不能让左右跨越对齐单元格

然后运行没有错误

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Beamer Presentation
% LaTeX Template
% Version 2.0 (March 8, 2022)
%
% This template originates from:
% https://www.LaTeXTemplates.com
%
% Author:
% Vel ([email protected])
%
% License:
% CC BY-NC-SA 4.0 (https://creativecommons.org/licenses/by-nc-sa/4.0/)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%----------------------------------------------------------------------------------------
%   PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------

\documentclass[
    11pt, % Set the default font size, options include: 8pt, 9pt, 10pt, 11pt, 12pt, 14pt, 17pt, 20pt
    %t, % Uncomment to vertically align all slide content to the top of the slide, rather than the default centered
    %aspectratio=169, % Uncomment to set the aspect ratio to a 16:9 ratio which matches the aspect ratio of 1080p and 4K screens and projectors
]{beamer}

\usepackage{dsfont}
\newcommand\mathhd{\mathds}%??


\graphicspath{{Images/}{./}} % Specifies where to look for included images (trailing slash required)

\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule for better rules in tables
% \usepackage{media9}
\usepackage{graphicx}
\usepackage{float}
\usepackage{algorithmicx}
\usepackage{algorithm}% http://ctan.org/pkg/algorithms
\usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx

%----------------------------------------------------------------------------------------
%   SELECT LAYOUT THEME
%----------------------------------------------------------------------------------------

% Beamer comes with a number of default layout themes which change the colors and layouts of slides. Below is a list of all themes available, uncomment each in turn to see what they look like.

%\usetheme{default}
%\usetheme{AnnArbor}
%\usetheme{Antibes}
%\usetheme{Bergen}
%\usetheme{Berkeley}
%\usetheme{Berlin}
%\usetheme{Boadilla}
%\usetheme{CambridgeUS}
%\usetheme{Copenhagen}
%\usetheme{Darmstadt}
%\usetheme{Dresden}
%\usetheme{Frankfurt}
%\usetheme{Goettingen}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usetheme{JuanLesPins}
%\usetheme{Luebeck}
\usetheme{Madrid}
%\usetheme{Malmoe}
%\usetheme{Marburg}
%\usetheme{Montpellier}
%\usetheme{PaloAlto}
%\usetheme{Pittsburgh}
%\usetheme{Rochester}
%\usetheme{Singapore}
%\usetheme{Szeged}
%\usetheme{Warsaw}

%----------------------------------------------------------------------------------------
%   SELECT COLOR THEME
%----------------------------------------------------------------------------------------

% Beamer comes with a number of color themes that can be applied to any layout theme to change its colors. Uncomment each of these in turn to see how they change the colors of your selected layout theme.

%\usecolortheme{albatross}
%\usecolortheme{beaver}
%\usecolortheme{beetle}
%\usecolortheme{crane}
%\usecolortheme{dolphin}
%\usecolortheme{dove}
%\usecolortheme{fly}
%\usecolortheme{lily}
%\usecolortheme{monarca}
%\usecolortheme{seagull}
%\usecolortheme{seahorse}
%\usecolortheme{spruce}
%\usecolortheme{whale}
%\usecolortheme{wolverine}

%----------------------------------------------------------------------------------------
%   SELECT FONT THEME & FONTS
%----------------------------------------------------------------------------------------

% Beamer comes with several font themes to easily change the fonts used in various parts of the presentation. Review the comments beside each one to decide if you would like to use it. Note that additional options can be specified for several of these font themes, consult the beamer documentation for more information.

\usefonttheme[onlymath]{serif} % Typeset using the default sans serif font
%\usefonttheme{serif} % Typeset using the default serif font (make sure a sans font isn't being set as the default font if you use this option!)
%\usefonttheme{structurebold} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in bold
%\usefonttheme{structureitalicserif} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in italic serif
%\usefonttheme{structuresmallcapsserif} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in small caps serif

%------------------------------------------------

%\usepackage{mathptmx} % Use the Times font for serif text
\usepackage{palatino} % Use the Palatino font for serif text

%\usepackage{helvet} % Use the Helvetica font for sans serif text
\usepackage[default]{opensans} % Use the Open Sans font for sans serif text
%\usepackage[default]{FiraSans} % Use the Fira Sans font for sans serif text
%\usepackage[default]{lato} % Use the Lato font for sans serif text

%----------------------------------------------------------------------------------------
%   SELECT INNER THEME
%----------------------------------------------------------------------------------------

% Inner themes change the styling of internal slide elements, for example: bullet points, blocks, bibliography entries, title pages, theorems, etc. Uncomment each theme in turn to see what changes it makes to your presentation.

%\useinnertheme{default}
\useinnertheme{circles}
%\useinnertheme{rectangles}
%\useinnertheme{rounded}
%\useinnertheme{inmargin}

%----------------------------------------------------------------------------------------
%   SELECT OUTER THEME
%----------------------------------------------------------------------------------------

% Outer themes change the overall layout of slides, such as: header and footer lines, sidebars and slide titles. Uncomment each theme in turn to see what changes it makes to your presentation.

%\useoutertheme{default}
%\useoutertheme{infolines}
%\useoutertheme{miniframes}
%\useoutertheme{smoothbars}
%\useoutertheme{sidebar}
%\useoutertheme{split}
%\useoutertheme{shadow}
%\useoutertheme{tree}
%\useoutertheme{smoothtree}

%\setbeamertemplate{footline} % Uncomment this line to remove the footer line in all slides
%\setbeamertemplate{footline}[page number] % Uncomment this line to replace the footer line in all slides with a simple slide count

%\setbeamertemplate{navigation symbols}{} % Uncomment this line to remove the navigation symbols from the bottom of all slides

%----------------------------------------------------------------------------------------
%   PRESENTATION INFORMATION
%----------------------------------------------------------------------------------------

\title[Mathematical Epidemiology Models]{Neural Network Solvers in Epidemiology Models} % The short title in the optional parameter appears at the bottom of every slide, the full title in the main parameter is only on the title page

\subtitle{A Prediction Model for Covid-19 Cases in Turkey} % Presentation subtitle, remove this command if a subtitle isn't required

\author[Günel \and Ahmad]{Korhan Günel \inst{1} \and Muhammad Jalil Ahmad \inst{2}}
% Affiliations
\institute[ADÜ \and UMBC]{\inst{1} Aydin Adnan Menderes University, Turkey\and \inst{2} University of Maryland, Baltimore Country, USA} % Presenter name(s), the optional parameter can contain a shortened version to appear on the bottom of every slide, while the main parameter will appear on the title slide

%\institute[]{University of Maryland, Baltimore County \\ \smallskip \textit{[email protected]}} % Your institution, the optional parameter can be used for the institution shorthand and will appear on the bottom of every slide after author names, while the required parameter is used on the title slide and can include your email address or additional information on separate lines

\date[November 11–13, 2022]{The 2022 International BEER Symposium \\ November 11–13, 2022} % Presentation date or conference/meeting name, the optional parameter can contain a shortened version to appear on the bottom of every slide, while the required parameter value is output to the title slide

%----------------------------------------------------------------------------------------

\begin{document}

%----------------------------------------------------------------------------------------
%   TITLE SLIDE
%----------------------------------------------------------------------------------------

\begin{frame}
    \titlepage % Output the title slide, automatically created using the text entered in the PRESENTATION INFORMATION block above
\end{frame}

%----------------------------------------------------------------------------------------
%   TABLE OF CONTENTS SLIDE
%----------------------------------------------------------------------------------------

% The table of contents outputs the sections and subsections that appear in your presentation, specified with the standard \section and \subsection commands. You may either display all sections and subsections on one slide with \tableofcontents, or display each section at a time on subsequent slides with \tableofcontents[pausesections]. The latter is useful if you want to step through each section and mention what you will discuss.

%\begin{frame}
%   \frametitle{Presentation Overview} % Slide title, remove this %command for no title
    
%   \tableofcontents % Output the table of contents (all sections on %one slide)
    %\tableofcontents[pausesections] % Output the table of contents %(break sections up across separate slides)
%\end{frame}

%----------------------------------------------------------------------------------------
%   PRESENTATION BODY SLIDES
%----------------------------------------------------------------------------------------


\subsection{Epidemiology Models}

\begin{frame}
    \frametitle{Susceptible-Infected ($SI$) Model}
        \begin{itemize}
        \item $S(t)$ represents number of people susceptible to disease at time $t$.
        \item $I(t)$ represents number of people already infected at time $t$.
        \item Total number of people in the population is given by $$N=S(t)+I(t)$$
\end{itemize}               
\begin{center}
%\includegraphics[scale=0.75]{SI model}
\end{center}
    
        
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected ($SI$) Model}
    \begin{itemize}
    
     \item Then the $SI$ model can be represented by the following system of ODEs:
        $$\dfrac{dS(t)}{dt} = -\dfrac{\beta S(t) I(t)}{N} $$
        $$\dfrac{dI(t)}{dt} = \dfrac{\beta S(t) I(t)}{N} $$
        with initial values, $S(0)=s_0>0$ and $I(0)=i_0>0$\\
    \item $\beta$ represents the rate at which the disease is spreading.
    \item Analytic solution of this system is give by:
        $$I(t) = \dfrac{i_0}{i_0+(1-i_0)e^{-\beta t}}
        $$  $$S(t)=N-I(t)$$
        \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
    \begin{itemize}
    \item In this model we will consider the case in which infected people become susceptible again after recovering from the disease.
    \begin{center}
%    \includegraphics[scale=0.75]{SIS model}
    \end{center}
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
    \begin{itemize}
    
    \item $SIS$ model is given by:
    $$\dfrac{dS(t)}{dt} = -\beta S(t) I(t)+\gamma I(t) $$
        $$\dfrac{dI(t)}{dt} = \beta S(t) I(t)-\gamma I(t) $$
        where $\gamma$ represents the rate at which infected people recover and become susceptible again.
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
    \begin{itemize}
    \item Analytic solution of this system is given by:
        $$I(t) = \dfrac{\alpha}{\beta+ C \alpha e^{-\alpha t}}$$  
        $$S(t)=N-\dfrac{\alpha}{\beta+ C \alpha e^{-\alpha t}}$$
        where $\alpha=\beta N-\gamma$ and $C=\dfrac{\alpha-i_0 \beta}{\alpha i_0}$
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Recovered ($SIR$) Model}
    \begin{itemize}
    
    \item $SIR$ model is given by:
    $$\dfrac{dS(t)}{dt} = -\beta S(t) I(t) $$
        $$\dfrac{dI(t)}{dt} = \beta S(t) I(t)-\gamma I(t) $$
        $$\dfrac{dR(t)}{dt} = \gamma I(t) $$
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIR$) Model}
    \begin{itemize}
    \item Analytic solution of this system is given by:
        $$S(u)=s_0 u$$  
        $$I(u)= \dfrac{\gamma}{\beta} \log u - \alpha u - \dfrac{C_1}{\beta}$$
        $$R(u)=-\dfrac{\gamma}{\beta} \log u$$
        where $C_1$ is an integration constant and $$t-t_0 = \int_{u_0}^u \dfrac{d\epsilon}{\epsilon (C_1-\gamma \log \epsilon + \alpha \beta u)}$$
    \end{itemize}
\end{frame}

%------------------------------------------------


\begin{algorithm}
\caption{MADS Algorithm}\label{alg:mABC}
\small
Given $f:\mathhd{R}^n \rightarrow \mathds{R}$ ve $x^0 \in \mathhd{R}^n$ is the starting point
\begin{algorithmic}[1]
\State Initialisation 
\State $\Delta^0  \in (0,\infty)$  \Comment{ initial frame size parameter }
\State $D = GZ $\Comment{ positive spanning matrix }
\State $\tau  \in (0,1), \: \tau  \in  \mathbb{Q} $\Comment{ mesh size adjustment parameter }
\State $\epsilon_{stop} \in [0, \infty) $\Comment{ stopping tolerance }
\State $k \gets 0$\Comment{ iteration counter }
\State Parameter Update
\State set the mesh size parameter to $\delta^k = min\lbrace \Delta^k , {(\Delta^k)}^2\rbrace $ 
\State Search
\State if $f(t) < f(x^k)$ for some $t$ in a finite subset $S^k$ of $M^k$ set $x^{k+1} \gets t$ and $\delta^{k+1} \gets \tau^{-1} \delta^k$ and go to 14 otherwise go to 11 
\algstore{bkbreak}
\end{algorithmic}
\end{algorithm}


%------------------------------------------------

\begin{algorithm}[h]
\begin{algorithmic}
\algrestore{bkbreak}
\State Poll
\State select a positive spanning set $\mathds{D}^k_\Delta$ such that $ P^k = \lbrace x^k + \delta^k d : d \in \mathds{D}^k_\Delta \rbrace $ is a subset of the frame $F^k$ of some extent $\Delta^k$.\\ if  $f(t) < f(x^k)$ for some $t \in P^k$ set $x^{k+1} \gets t$ ve $\delta^{k+1} \gets \tau^{-1} \delta^k$ otherwise set $x^{k+1} \gets x^k$ and $\delta^{k+1} \gets \tau^{k} \delta^k$.

\State Termination
\State If $\Delta^{k+1} \geq \epsilon_{stop}$ increment $k \gets k+1$ and go to 9 otherwise stop. 
\end{algorithmic}
\end{algorithm}

%------------------------------------------------

\begin{frame}
    \frametitle{2-D Example for working of MADS Algorithm}
    \begin{center}
%    \includegraphics[scale=0.5]{Frame 1}
    \end{center}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{2-D Example for working of MADS Algorithm}
    \begin{center}
%    \includegraphics[scale=0.5]{Frame 2}
    \end{center}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Artificial Neural Networks}
    \begin{itemize}
    \item Single Layers Neural Nets
    \begin{itemize}
    \item For linear functions
    \end{itemize}
    \item Multi-Layers Neural Nets
    \begin{itemize}
    \item For non-linear functions
    \end{itemize}
    \end{itemize}
    \begin{center}
    \begin{figure}
%    \includegraphics[scale=0.5]{NeuralNet}
    \caption{Topology of Neural Network}
    \end{figure}
    \end{center}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Neural Network solvers for $SI$ Model}
    $$S_t=s_0+(t-t_0 ) Net(t_j,s_j,i_j,\overrightarrow{p_S}) $$
    $$T_t=i_0+(t-t_0 ) Net(t_j,s_j,i_j,\overrightarrow{p_I}) $$
    \begin{itemize}
    \item $h>0$ is step size
    \item $t_j=t_0+j \cdot h $ for $j=0,1, \ldots ,n$ gives us discretization of the interval $[t_0, t_n]$
    \item $\overrightarrow{p_S}=\overrightarrow{p_S}(\overrightarrow{\alpha_j},\overrightarrow{\omega_j},\overrightarrow{\beta_j})$ and $\overrightarrow{p_I}=\overrightarrow{p_I}(\overrightarrow{\alpha_j},\overrightarrow{\omega_j},\overrightarrow{\beta_j})$ are unknown parameters of the neural network  
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Neural Network solvers for $SI$ Model}
    \begin{itemize}
    \item Points $t_j$ are used to train neural network
    \item The output of the NN is given by $$Net(t_j, s_j, i_j, \overrightarrow{p_S})= \sum_{i=1}^{m_1} \alpha_{S i} \ \sigma(z_{1i}) $$
    \item $\sigma(z) = \dfrac{1}{1+e^{-z}} $ is the activation function
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}

$$\dfrac{\partial S_t}{\partial t} = Net(t_j,s_j,i_j,\overrightarrow{p_S})+(t-t_0) \dfrac{\partial Net(t_j,s_j,i_j,\overrightarrow{p_S})}{\partial t} $$

$$\dfrac{\partial I_t}{\partial t} = Net(t_j,s_j,i_j,\overrightarrow{p_I})+(t-t_0) \dfrac{\partial Net(t_j,s_j,i_j,\overrightarrow{p_I})}{\partial t} $$

$$\dfrac{dS(t)}{dt}=f(t, S(t),I(t)) $$

$$\dfrac{dI(t)}{dt}=g(t, S(t),I(t)) $$

\end{frame}

%------------------------------------------------


\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}

Cost function for $SI$ model is given by:
$$E= \dfrac{1}{2n} \sum_{i=1}^n \left\lbrace \left[ \dfrac{\partial S(t_i)}{\partial t_i}-f(t_i, S(t_i),I(t_i))\right]^2 +  \left[ \dfrac{\partial I(t_i)}{\partial t_i}-g(t_i, S(t_i),I(t_i))\right]^2\right\rbrace   $$



\end{frame}

%------------------------------------------------


\begin{frame}
    \frametitle{Cost function for $SIR$}
    Similarly cost function for $SIR$ model is given by:
\begin{multline*} E= \dfrac{1}{2n} \sum_{i=1}^n \left\lbrace \left[ \dfrac{\partial S(t_i)}{\partial t_i}-f(t_i, S(t_i),I(t_i),R(t_i))\right]^2 +  \right.\\ 
\left.\left[ \dfrac{\partial I(t_i)}{\partial t_i}-g(t_i, S(t_i),I(t_i),R(t_i))\right]^2 + \left[ \dfrac{\partial R(t_i)}{\partial t_i}-h(t_i, S(t_i),I(t_i),R(t_i))\right]^2\right\rbrace   
\end{multline*}
\end{frame}

%------------------------------------------------


\begin{frame}
    \frametitle{Error Calculation}
    Absolute error for the models is given by:
$$E(S)=|s_j - S_t(t_j)| $$
$$E(I)=|i_j - I_t(t_j)| $$
$$E(R)=|r_j - R_t(t_j)| $$
\end{frame}

%------------------------------------------------

\subsection{Columns}

\begin{frame}
    \frametitle{Multiple Columns}
    \framesubtitle{Subtitle} % Optional subtitle
    
    \begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
        \begin{column}{0.45\textwidth} % Left column width
            \textbf{Heading}
            \begin{enumerate}
                \item Statement
                \item Explanation
                \item Example
            \end{enumerate}
        \end{column}
        \begin{column}{0.5\textwidth} % Right column width
            Lorem ipsum dolor sit amet, consectetur adipiscing elit. Integer lectus nisl, ultricies in feugiat rutrum, porttitor sit amet augue. Aliquam ut tortor mauris. Sed volutpat ante purus, quis accumsan dolor.
        \end{column}
    \end{columns}
\end{frame}

%------------------------------------------------

\section{Table and Figure Examples}

\subsection{Table}

\begin{frame}
    \frametitle{Table}
    \framesubtitle{Subtitle} % Optional subtitle
    
    \begin{table}
        \begin{tabular}{l l l}
            \toprule
            \textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2}\\
            \midrule
            Treatment 1 & 0.0003262 & 0.562 \\
            Treatment 2 & 0.0015681 & 0.910 \\
            Treatment 3 & 0.0009271 & 0.296 \\
            \bottomrule
        \end{tabular}
        \caption{Table caption}
    \end{table}
\end{frame}

%------------------------------------------------

\subsection{Figure}

\begin{frame}
    \frametitle{Figure}
    
    \begin{figure}
%        \includegraphics[width=0.8\linewidth]{creodocs_logo.pdf}
        \caption{Creodocs logo.}
    \end{figure}
\end{frame}

%------------------------------------------------

\section{Mathematics}

\begin{frame}
    \frametitle{Definitions \& Examples}
    
    \begin{definition}
        A \alert{prime number} is a number that has exactly two divisors.
    \end{definition}
    
    \smallskip % Vertical whitespace
    
    \begin{example}
        \begin{itemize}
            \item 2 is prime (two divisors: 1 and 2).
            \item 3 is prime (two divisors: 1 and 3).
            \item 4 is not prime (\alert{three} divisors: 1, 2, and 4).
        \end{itemize}
    \end{example}
    
    \smallskip % Vertical whitespace
    
    You can also use the \texttt{theorem}, \texttt{lemma}, \texttt{proof} and \texttt{corollary} environments.
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Theorem, Corollary \& Proof}
    
    \begin{theorem}[Mass--energy equivalence]
        $E = mc^2$
    \end{theorem}
    
    \begin{corollary}
        $x + y = y + x$
    \end{corollary}
    
    \begin{proof}
        $\omega + \phi = \epsilon$
    \end{proof}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Equation}

    \begin{equation}
        \cos^3 \theta =\frac{1}{4}\cos\theta+\frac{3}{4}\cos 3\theta
    \end{equation}
\end{frame}

%------------------------------------------------

\begin{frame}[fragile] % Need to use the fragile option when verbatim is used in the slide
    \frametitle{Verbatim}
    
    \begin{example}[Theorem Slide Code]
        \begin{verbatim}
            \begin{frame}
                \frametitle{Theorem}
                \begin{theorem}[Mass--energy equivalence]
                    $E = mc^2$
                \end{theorem}
        \end{frame}\end{verbatim} % Must be on the same line
    \end{example}
\end{frame}

%------------------------------------------------

\begin{frame}
    Slide without title.
\end{frame}

%------------------------------------------------

\section{Referencing}

\begin{frame}
    \frametitle{Citing References}
    
    An example of the \texttt{\textbackslash cite} command to cite within the presentation:
    
    \bigskip % Vertical whitespace
    
    This statement requires citation \cite{p1,p2}.
\end{frame}

%------------------------------------------------

\begin{frame} % Use [allowframebreaks] to allow automatic splitting across slides if the content is too long
    \frametitle{References}
    
    \begin{thebibliography}{99} % Beamer does not support BibTeX so references must be inserted manually as below, you may need to use multiple columns and/or reduce the font size further if you have many references
        \footnotesize % Reduce the font size in the bibliography
        
        \bibitem[Smith, 2022]{p1}
            John Smith (2022)
            \newblock Publication title
            \newblock \emph{Journal Name} 12(3), 45 -- 678.
            
        \bibitem[Kennedy, 2023]{p2}
            Annabelle Kennedy (2023)
            \newblock Publication title
            \newblock \emph{Journal Name} 12(3), 45 -- 678.
    \end{thebibliography}
\end{frame}

%----------------------------------------------------------------------------------------
%   ACKNOWLEDGMENTS SLIDE
%----------------------------------------------------------------------------------------

\begin{frame}
    \frametitle{Acknowledgements}
    
    \begin{columns}[t] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
        \begin{column}{0.45\textwidth} % Left column width
            \textbf{Smith Lab}
            \begin{itemize}
                \item Alice Smith
                \item Devon Brown
            \end{itemize}
            \textbf{Cook Lab}
            \begin{itemize}
                \item Margaret
                \item Jennifer
                \item Yuan
            \end{itemize}
        \end{column}        
        \begin{column}{0.5\textwidth} % Right column width
            \textbf{Funding}
            \begin{itemize}
                \item British Royal Navy
                \item Norwegian Government
            \end{itemize}
        \end{column}
    \end{columns}
\end{frame}

%----------------------------------------------------------------------------------------
%   CLOSING SLIDE
%----------------------------------------------------------------------------------------

\begin{frame}[plain] % The optional argument 'plain' hides the headline and footline
    \begin{center}
        {\Huge The End}
        
        \bigskip\bigskip % Vertical whitespace
        
        {\LARGE Questions? Comments?}
    \end{center}
\end{frame}

%----------------------------------------------------------------------------------------

\end{document}

Answer 1

您没有从日志中显示错误格式。

忽略丢失的图像，第一个错误是

! Undefined control sequence.
l.318 ... \lbrace x^k + \delta^k d : d \in \mathds
                                                  {D}^k_\Delta \rbrace $ is ...

?

添加

\usepackage{dsfont}
\newcommand\mathhd{\mathds}%??

然后你得到

! Double superscript.
l.322 ...^k$ and $\delta^{k+1} \gets \tau^ \delta^
                                                  k$.
?

因为你不能拥有x^2^3 我猜你的意思是\tau^{k}？

然后

! LaTeX Error: Unicode character   (U+205F)
               not set up for use with LaTeX.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.380 \end{frame}
                 
?

U+205F 是 Net 前面的空格，0 ) Net用空格代替

生产

! Missing \right. inserted.
<inserted text> 
                \right .
l.429 \end{frame}
                 
?

因为你不能让左右跨越对齐单元格

然后运行没有错误

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Beamer Presentation
% LaTeX Template
% Version 2.0 (March 8, 2022)
%
% This template originates from:
% https://www.LaTeXTemplates.com
%
% Author:
% Vel ([email protected])
%
% License:
% CC BY-NC-SA 4.0 (https://creativecommons.org/licenses/by-nc-sa/4.0/)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%----------------------------------------------------------------------------------------
%   PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------

\documentclass[
    11pt, % Set the default font size, options include: 8pt, 9pt, 10pt, 11pt, 12pt, 14pt, 17pt, 20pt
    %t, % Uncomment to vertically align all slide content to the top of the slide, rather than the default centered
    %aspectratio=169, % Uncomment to set the aspect ratio to a 16:9 ratio which matches the aspect ratio of 1080p and 4K screens and projectors
]{beamer}

\usepackage{dsfont}
\newcommand\mathhd{\mathds}%??


\graphicspath{{Images/}{./}} % Specifies where to look for included images (trailing slash required)

\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule for better rules in tables
% \usepackage{media9}
\usepackage{graphicx}
\usepackage{float}
\usepackage{algorithmicx}
\usepackage{algorithm}% http://ctan.org/pkg/algorithms
\usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx

%----------------------------------------------------------------------------------------
%   SELECT LAYOUT THEME
%----------------------------------------------------------------------------------------

% Beamer comes with a number of default layout themes which change the colors and layouts of slides. Below is a list of all themes available, uncomment each in turn to see what they look like.

%\usetheme{default}
%\usetheme{AnnArbor}
%\usetheme{Antibes}
%\usetheme{Bergen}
%\usetheme{Berkeley}
%\usetheme{Berlin}
%\usetheme{Boadilla}
%\usetheme{CambridgeUS}
%\usetheme{Copenhagen}
%\usetheme{Darmstadt}
%\usetheme{Dresden}
%\usetheme{Frankfurt}
%\usetheme{Goettingen}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usetheme{JuanLesPins}
%\usetheme{Luebeck}
\usetheme{Madrid}
%\usetheme{Malmoe}
%\usetheme{Marburg}
%\usetheme{Montpellier}
%\usetheme{PaloAlto}
%\usetheme{Pittsburgh}
%\usetheme{Rochester}
%\usetheme{Singapore}
%\usetheme{Szeged}
%\usetheme{Warsaw}

%----------------------------------------------------------------------------------------
%   SELECT COLOR THEME
%----------------------------------------------------------------------------------------

% Beamer comes with a number of color themes that can be applied to any layout theme to change its colors. Uncomment each of these in turn to see how they change the colors of your selected layout theme.

%\usecolortheme{albatross}
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%\usecolortheme{beetle}
%\usecolortheme{crane}
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%\usecolortheme{dove}
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%\usecolortheme{monarca}
%\usecolortheme{seagull}
%\usecolortheme{seahorse}
%\usecolortheme{spruce}
%\usecolortheme{whale}
%\usecolortheme{wolverine}

%----------------------------------------------------------------------------------------
%   SELECT FONT THEME & FONTS
%----------------------------------------------------------------------------------------

% Beamer comes with several font themes to easily change the fonts used in various parts of the presentation. Review the comments beside each one to decide if you would like to use it. Note that additional options can be specified for several of these font themes, consult the beamer documentation for more information.

\usefonttheme[onlymath]{serif} % Typeset using the default sans serif font
%\usefonttheme{serif} % Typeset using the default serif font (make sure a sans font isn't being set as the default font if you use this option!)
%\usefonttheme{structurebold} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in bold
%\usefonttheme{structureitalicserif} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in italic serif
%\usefonttheme{structuresmallcapsserif} % Typeset important structure text (titles, headlines, footlines, sidebar, etc) in small caps serif

%------------------------------------------------

%\usepackage{mathptmx} % Use the Times font for serif text
\usepackage{palatino} % Use the Palatino font for serif text

%\usepackage{helvet} % Use the Helvetica font for sans serif text
\usepackage[default]{opensans} % Use the Open Sans font for sans serif text
%\usepackage[default]{FiraSans} % Use the Fira Sans font for sans serif text
%\usepackage[default]{lato} % Use the Lato font for sans serif text

%----------------------------------------------------------------------------------------
%   SELECT INNER THEME
%----------------------------------------------------------------------------------------

% Inner themes change the styling of internal slide elements, for example: bullet points, blocks, bibliography entries, title pages, theorems, etc. Uncomment each theme in turn to see what changes it makes to your presentation.

%\useinnertheme{default}
\useinnertheme{circles}
%\useinnertheme{rectangles}
%\useinnertheme{rounded}
%\useinnertheme{inmargin}

%----------------------------------------------------------------------------------------
%   SELECT OUTER THEME
%----------------------------------------------------------------------------------------

% Outer themes change the overall layout of slides, such as: header and footer lines, sidebars and slide titles. Uncomment each theme in turn to see what changes it makes to your presentation.

%\useoutertheme{default}
%\useoutertheme{infolines}
%\useoutertheme{miniframes}
%\useoutertheme{smoothbars}
%\useoutertheme{sidebar}
%\useoutertheme{split}
%\useoutertheme{shadow}
%\useoutertheme{tree}
%\useoutertheme{smoothtree}

%\setbeamertemplate{footline} % Uncomment this line to remove the footer line in all slides
%\setbeamertemplate{footline}[page number] % Uncomment this line to replace the footer line in all slides with a simple slide count

%\setbeamertemplate{navigation symbols}{} % Uncomment this line to remove the navigation symbols from the bottom of all slides

%----------------------------------------------------------------------------------------
%   PRESENTATION INFORMATION
%----------------------------------------------------------------------------------------

\title[Mathematical Epidemiology Models]{Neural Network Solvers in Epidemiology Models} % The short title in the optional parameter appears at the bottom of every slide, the full title in the main parameter is only on the title page

\subtitle{A Prediction Model for Covid-19 Cases in Turkey} % Presentation subtitle, remove this command if a subtitle isn't required

\author[Günel \and Ahmad]{Korhan Günel \inst{1} \and Muhammad Jalil Ahmad \inst{2}}
% Affiliations
\institute[ADÜ \and UMBC]{\inst{1} Aydin Adnan Menderes University, Turkey\and \inst{2} University of Maryland, Baltimore Country, USA} % Presenter name(s), the optional parameter can contain a shortened version to appear on the bottom of every slide, while the main parameter will appear on the title slide

%\institute[]{University of Maryland, Baltimore County \\ \smallskip \textit{[email protected]}} % Your institution, the optional parameter can be used for the institution shorthand and will appear on the bottom of every slide after author names, while the required parameter is used on the title slide and can include your email address or additional information on separate lines

\date[November 11–13, 2022]{The 2022 International BEER Symposium \\ November 11–13, 2022} % Presentation date or conference/meeting name, the optional parameter can contain a shortened version to appear on the bottom of every slide, while the required parameter value is output to the title slide

%----------------------------------------------------------------------------------------

\begin{document}

%----------------------------------------------------------------------------------------
%   TITLE SLIDE
%----------------------------------------------------------------------------------------

\begin{frame}
    \titlepage % Output the title slide, automatically created using the text entered in the PRESENTATION INFORMATION block above
\end{frame}

%----------------------------------------------------------------------------------------
%   TABLE OF CONTENTS SLIDE
%----------------------------------------------------------------------------------------

% The table of contents outputs the sections and subsections that appear in your presentation, specified with the standard \section and \subsection commands. You may either display all sections and subsections on one slide with \tableofcontents, or display each section at a time on subsequent slides with \tableofcontents[pausesections]. The latter is useful if you want to step through each section and mention what you will discuss.

%\begin{frame}
%   \frametitle{Presentation Overview} % Slide title, remove this %command for no title
    
%   \tableofcontents % Output the table of contents (all sections on %one slide)
    %\tableofcontents[pausesections] % Output the table of contents %(break sections up across separate slides)
%\end{frame}

%----------------------------------------------------------------------------------------
%   PRESENTATION BODY SLIDES
%----------------------------------------------------------------------------------------


\subsection{Epidemiology Models}

\begin{frame}
    \frametitle{Susceptible-Infected ($SI$) Model}
        \begin{itemize}
        \item $S(t)$ represents number of people susceptible to disease at time $t$.
        \item $I(t)$ represents number of people already infected at time $t$.
        \item Total number of people in the population is given by $$N=S(t)+I(t)$$
\end{itemize}               
\begin{center}
%\includegraphics[scale=0.75]{SI model}
\end{center}
    
        
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected ($SI$) Model}
    \begin{itemize}
    
     \item Then the $SI$ model can be represented by the following system of ODEs:
        $$\dfrac{dS(t)}{dt} = -\dfrac{\beta S(t) I(t)}{N} $$
        $$\dfrac{dI(t)}{dt} = \dfrac{\beta S(t) I(t)}{N} $$
        with initial values, $S(0)=s_0>0$ and $I(0)=i_0>0$\\
    \item $\beta$ represents the rate at which the disease is spreading.
    \item Analytic solution of this system is give by:
        $$I(t) = \dfrac{i_0}{i_0+(1-i_0)e^{-\beta t}}
        $$  $$S(t)=N-I(t)$$
        \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
    \begin{itemize}
    \item In this model we will consider the case in which infected people become susceptible again after recovering from the disease.
    \begin{center}
%    \includegraphics[scale=0.75]{SIS model}
    \end{center}
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
    \begin{itemize}
    
    \item $SIS$ model is given by:
    $$\dfrac{dS(t)}{dt} = -\beta S(t) I(t)+\gamma I(t) $$
        $$\dfrac{dI(t)}{dt} = \beta S(t) I(t)-\gamma I(t) $$
        where $\gamma$ represents the rate at which infected people recover and become susceptible again.
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIS$) Model}
    \begin{itemize}
    \item Analytic solution of this system is given by:
        $$I(t) = \dfrac{\alpha}{\beta+ C \alpha e^{-\alpha t}}$$  
        $$S(t)=N-\dfrac{\alpha}{\beta+ C \alpha e^{-\alpha t}}$$
        where $\alpha=\beta N-\gamma$ and $C=\dfrac{\alpha-i_0 \beta}{\alpha i_0}$
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Recovered ($SIR$) Model}
    \begin{itemize}
    
    \item $SIR$ model is given by:
    $$\dfrac{dS(t)}{dt} = -\beta S(t) I(t) $$
        $$\dfrac{dI(t)}{dt} = \beta S(t) I(t)-\gamma I(t) $$
        $$\dfrac{dR(t)}{dt} = \gamma I(t) $$
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Susceptible-Infected-Susceptible ($SIR$) Model}
    \begin{itemize}
    \item Analytic solution of this system is given by:
        $$S(u)=s_0 u$$  
        $$I(u)= \dfrac{\gamma}{\beta} \log u - \alpha u - \dfrac{C_1}{\beta}$$
        $$R(u)=-\dfrac{\gamma}{\beta} \log u$$
        where $C_1$ is an integration constant and $$t-t_0 = \int_{u_0}^u \dfrac{d\epsilon}{\epsilon (C_1-\gamma \log \epsilon + \alpha \beta u)}$$
    \end{itemize}
\end{frame}

%------------------------------------------------


\begin{algorithm}
\caption{MADS Algorithm}\label{alg:mABC}
\small
Given $f:\mathhd{R}^n \rightarrow \mathds{R}$ ve $x^0 \in \mathhd{R}^n$ is the starting point
\begin{algorithmic}[1]
\State Initialisation 
\State $\Delta^0  \in (0,\infty)$  \Comment{ initial frame size parameter }
\State $D = GZ $\Comment{ positive spanning matrix }
\State $\tau  \in (0,1), \: \tau  \in  \mathbb{Q} $\Comment{ mesh size adjustment parameter }
\State $\epsilon_{stop} \in [0, \infty) $\Comment{ stopping tolerance }
\State $k \gets 0$\Comment{ iteration counter }
\State Parameter Update
\State set the mesh size parameter to $\delta^k = min\lbrace \Delta^k , {(\Delta^k)}^2\rbrace $ 
\State Search
\State if $f(t) < f(x^k)$ for some $t$ in a finite subset $S^k$ of $M^k$ set $x^{k+1} \gets t$ and $\delta^{k+1} \gets \tau^{-1} \delta^k$ and go to 14 otherwise go to 11 
\algstore{bkbreak}
\end{algorithmic}
\end{algorithm}


%------------------------------------------------

\begin{algorithm}[h]
\begin{algorithmic}
\algrestore{bkbreak}
\State Poll
\State select a positive spanning set $\mathds{D}^k_\Delta$ such that $ P^k = \lbrace x^k + \delta^k d : d \in \mathds{D}^k_\Delta \rbrace $ is a subset of the frame $F^k$ of some extent $\Delta^k$.\\ if  $f(t) < f(x^k)$ for some $t \in P^k$ set $x^{k+1} \gets t$ ve $\delta^{k+1} \gets \tau^{-1} \delta^k$ otherwise set $x^{k+1} \gets x^k$ and $\delta^{k+1} \gets \tau^{k} \delta^k$.

\State Termination
\State If $\Delta^{k+1} \geq \epsilon_{stop}$ increment $k \gets k+1$ and go to 9 otherwise stop. 
\end{algorithmic}
\end{algorithm}

%------------------------------------------------

\begin{frame}
    \frametitle{2-D Example for working of MADS Algorithm}
    \begin{center}
%    \includegraphics[scale=0.5]{Frame 1}
    \end{center}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{2-D Example for working of MADS Algorithm}
    \begin{center}
%    \includegraphics[scale=0.5]{Frame 2}
    \end{center}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Artificial Neural Networks}
    \begin{itemize}
    \item Single Layers Neural Nets
    \begin{itemize}
    \item For linear functions
    \end{itemize}
    \item Multi-Layers Neural Nets
    \begin{itemize}
    \item For non-linear functions
    \end{itemize}
    \end{itemize}
    \begin{center}
    \begin{figure}
%    \includegraphics[scale=0.5]{NeuralNet}
    \caption{Topology of Neural Network}
    \end{figure}
    \end{center}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Neural Network solvers for $SI$ Model}
    $$S_t=s_0+(t-t_0 ) Net(t_j,s_j,i_j,\overrightarrow{p_S}) $$
    $$T_t=i_0+(t-t_0 ) Net(t_j,s_j,i_j,\overrightarrow{p_I}) $$
    \begin{itemize}
    \item $h>0$ is step size
    \item $t_j=t_0+j \cdot h $ for $j=0,1, \ldots ,n$ gives us discretization of the interval $[t_0, t_n]$
    \item $\overrightarrow{p_S}=\overrightarrow{p_S}(\overrightarrow{\alpha_j},\overrightarrow{\omega_j},\overrightarrow{\beta_j})$ and $\overrightarrow{p_I}=\overrightarrow{p_I}(\overrightarrow{\alpha_j},\overrightarrow{\omega_j},\overrightarrow{\beta_j})$ are unknown parameters of the neural network  
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Neural Network solvers for $SI$ Model}
    \begin{itemize}
    \item Points $t_j$ are used to train neural network
    \item The output of the NN is given by $$Net(t_j, s_j, i_j, \overrightarrow{p_S})= \sum_{i=1}^{m_1} \alpha_{S i} \ \sigma(z_{1i}) $$
    \item $\sigma(z) = \dfrac{1}{1+e^{-z}} $ is the activation function
    \end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}

$$\dfrac{\partial S_t}{\partial t} = Net(t_j,s_j,i_j,\overrightarrow{p_S})+(t-t_0) \dfrac{\partial Net(t_j,s_j,i_j,\overrightarrow{p_S})}{\partial t} $$

$$\dfrac{\partial I_t}{\partial t} = Net(t_j,s_j,i_j,\overrightarrow{p_I})+(t-t_0) \dfrac{\partial Net(t_j,s_j,i_j,\overrightarrow{p_I})}{\partial t} $$

$$\dfrac{dS(t)}{dt}=f(t, S(t),I(t)) $$

$$\dfrac{dI(t)}{dt}=g(t, S(t),I(t)) $$

\end{frame}

%------------------------------------------------


\begin{frame}
\frametitle{Neural Network solvers for $SI$ Model}

Cost function for $SI$ model is given by:
$$E= \dfrac{1}{2n} \sum_{i=1}^n \left\lbrace \left[ \dfrac{\partial S(t_i)}{\partial t_i}-f(t_i, S(t_i),I(t_i))\right]^2 +  \left[ \dfrac{\partial I(t_i)}{\partial t_i}-g(t_i, S(t_i),I(t_i))\right]^2\right\rbrace   $$



\end{frame}

%------------------------------------------------


\begin{frame}
    \frametitle{Cost function for $SIR$}
    Similarly cost function for $SIR$ model is given by:
\begin{multline*} E= \dfrac{1}{2n} \sum_{i=1}^n \left\lbrace \left[ \dfrac{\partial S(t_i)}{\partial t_i}-f(t_i, S(t_i),I(t_i),R(t_i))\right]^2 +  \right.\\ 
\left.\left[ \dfrac{\partial I(t_i)}{\partial t_i}-g(t_i, S(t_i),I(t_i),R(t_i))\right]^2 + \left[ \dfrac{\partial R(t_i)}{\partial t_i}-h(t_i, S(t_i),I(t_i),R(t_i))\right]^2\right\rbrace   
\end{multline*}
\end{frame}

%------------------------------------------------


\begin{frame}
    \frametitle{Error Calculation}
    Absolute error for the models is given by:
$$E(S)=|s_j - S_t(t_j)| $$
$$E(I)=|i_j - I_t(t_j)| $$
$$E(R)=|r_j - R_t(t_j)| $$
\end{frame}

%------------------------------------------------

\subsection{Columns}

\begin{frame}
    \frametitle{Multiple Columns}
    \framesubtitle{Subtitle} % Optional subtitle
    
    \begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
        \begin{column}{0.45\textwidth} % Left column width
            \textbf{Heading}
            \begin{enumerate}
                \item Statement
                \item Explanation
                \item Example
            \end{enumerate}
        \end{column}
        \begin{column}{0.5\textwidth} % Right column width
            Lorem ipsum dolor sit amet, consectetur adipiscing elit. Integer lectus nisl, ultricies in feugiat rutrum, porttitor sit amet augue. Aliquam ut tortor mauris. Sed volutpat ante purus, quis accumsan dolor.
        \end{column}
    \end{columns}
\end{frame}

%------------------------------------------------

\section{Table and Figure Examples}

\subsection{Table}

\begin{frame}
    \frametitle{Table}
    \framesubtitle{Subtitle} % Optional subtitle
    
    \begin{table}
        \begin{tabular}{l l l}
            \toprule
            \textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2}\\
            \midrule
            Treatment 1 & 0.0003262 & 0.562 \\
            Treatment 2 & 0.0015681 & 0.910 \\
            Treatment 3 & 0.0009271 & 0.296 \\
            \bottomrule
        \end{tabular}
        \caption{Table caption}
    \end{table}
\end{frame}

%------------------------------------------------

\subsection{Figure}

\begin{frame}
    \frametitle{Figure}
    
    \begin{figure}
%        \includegraphics[width=0.8\linewidth]{creodocs_logo.pdf}
        \caption{Creodocs logo.}
    \end{figure}
\end{frame}

%------------------------------------------------

\section{Mathematics}

\begin{frame}
    \frametitle{Definitions \& Examples}
    
    \begin{definition}
        A \alert{prime number} is a number that has exactly two divisors.
    \end{definition}
    
    \smallskip % Vertical whitespace
    
    \begin{example}
        \begin{itemize}
            \item 2 is prime (two divisors: 1 and 2).
            \item 3 is prime (two divisors: 1 and 3).
            \item 4 is not prime (\alert{three} divisors: 1, 2, and 4).
        \end{itemize}
    \end{example}
    
    \smallskip % Vertical whitespace
    
    You can also use the \texttt{theorem}, \texttt{lemma}, \texttt{proof} and \texttt{corollary} environments.
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Theorem, Corollary \& Proof}
    
    \begin{theorem}[Mass--energy equivalence]
        $E = mc^2$
    \end{theorem}
    
    \begin{corollary}
        $x + y = y + x$
    \end{corollary}
    
    \begin{proof}
        $\omega + \phi = \epsilon$
    \end{proof}
\end{frame}

%------------------------------------------------

\begin{frame}
    \frametitle{Equation}

    \begin{equation}
        \cos^3 \theta =\frac{1}{4}\cos\theta+\frac{3}{4}\cos 3\theta
    \end{equation}
\end{frame}

%------------------------------------------------

\begin{frame}[fragile] % Need to use the fragile option when verbatim is used in the slide
    \frametitle{Verbatim}
    
    \begin{example}[Theorem Slide Code]
        \begin{verbatim}
            \begin{frame}
                \frametitle{Theorem}
                \begin{theorem}[Mass--energy equivalence]
                    $E = mc^2$
                \end{theorem}
        \end{frame}\end{verbatim} % Must be on the same line
    \end{example}
\end{frame}

%------------------------------------------------

\begin{frame}
    Slide without title.
\end{frame}

%------------------------------------------------

\section{Referencing}

\begin{frame}
    \frametitle{Citing References}
    
    An example of the \texttt{\textbackslash cite} command to cite within the presentation:
    
    \bigskip % Vertical whitespace
    
    This statement requires citation \cite{p1,p2}.
\end{frame}

%------------------------------------------------

\begin{frame} % Use [allowframebreaks] to allow automatic splitting across slides if the content is too long
    \frametitle{References}
    
    \begin{thebibliography}{99} % Beamer does not support BibTeX so references must be inserted manually as below, you may need to use multiple columns and/or reduce the font size further if you have many references
        \footnotesize % Reduce the font size in the bibliography
        
        \bibitem[Smith, 2022]{p1}
            John Smith (2022)
            \newblock Publication title
            \newblock \emph{Journal Name} 12(3), 45 -- 678.
            
        \bibitem[Kennedy, 2023]{p2}
            Annabelle Kennedy (2023)
            \newblock Publication title
            \newblock \emph{Journal Name} 12(3), 45 -- 678.
    \end{thebibliography}
\end{frame}

%----------------------------------------------------------------------------------------
%   ACKNOWLEDGMENTS SLIDE
%----------------------------------------------------------------------------------------

\begin{frame}
    \frametitle{Acknowledgements}
    
    \begin{columns}[t] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
        \begin{column}{0.45\textwidth} % Left column width
            \textbf{Smith Lab}
            \begin{itemize}
                \item Alice Smith
                \item Devon Brown
            \end{itemize}
            \textbf{Cook Lab}
            \begin{itemize}
                \item Margaret
                \item Jennifer
                \item Yuan
            \end{itemize}
        \end{column}        
        \begin{column}{0.5\textwidth} % Right column width
            \textbf{Funding}
            \begin{itemize}
                \item British Royal Navy
                \item Norwegian Government
            \end{itemize}
        \end{column}
    \end{columns}
\end{frame}

%----------------------------------------------------------------------------------------
%   CLOSING SLIDE
%----------------------------------------------------------------------------------------

\begin{frame}[plain] % The optional argument 'plain' hides the headline and footline
    \begin{center}
        {\Huge The End}
        
        \bigskip\bigskip % Vertical whitespace
        
        {\LARGE Questions? Comments?}
    \end{center}
\end{frame}

%----------------------------------------------------------------------------------------

\end{document}

在 BEAMER 中出现以下错误

答案1

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