\section{Appendix}

\subsection{One-dimensional non-degeneracy problem}

As for the one-dimensional problem, we mostly consider the possibility of solving multiple non-degenerate eigenstates and eigenvalues simultaneously, as well as the influence of hyperparameters and eigenvalue forms on convergence. The one-dimensional infinite deep potential is representative of the finite margin problem, and the one-dimensional harmonic oscillator is representative of the infinite margin problem, and both have precise solutions. This also opens up the possibility for comprehensive error analysis. The two problems are analyzed as follows Table.\ref{table.problem1d}.

\begin{table}[H]\centering
  \caption{one-dimensional problem, where $H_n(x)$ is Hermite polynomial,and $n\in \mathbb{N} $ is energy level.}

  \label{table.problem1d}

  \begin{tabular}{lllllll}

  \toprule

    Symbol& 3D Harmonic Oscillator & 2D Infinite Well&Illustrate \\

  \midrule

  $V(x)$ &${1\over 2} x^2$ & $0,x\in{other}; \infty,x\in{(0,1)}$&Potential\\

  $\mathcal{B} c$ &$\Psi(x)\stackrel{x\rightarrow \pm\infty}{\longrightarrow}0$ &$\Psi(x)\stackrel{x=0,1}{\longrightarrow}0$&Boundary condition\\

  $\mathcal{B}(x)$ &$e^{-{1\over2}x^2}$ & $x(1-x)$&Boundary Term\\

  $\Psi^R_n$ &$N_nH_n(x)e^{-{x^2\over2}}$ & $\sqrt{2}sin(n\pi x)$&Real Wave Function\\

  $E^R_n$ &$n+{1\over 2}$ & ${{n^2\pi^2}\over2}$&Real Energy\\

  \bottomrule

  \end{tabular}

  \end{table}

To facilitate convergence, the sampling method and the eigenvalue form need to be redesigned for the problem, as shown in the following Table.\ref{table.term1d}.

  \begin{table}[H]\centering

    \caption{Key indicators of one-dimensional potential training, where $\lambda$ is the parameter to be trained.pdf=1 indicates uniform sampling within the domain}
    \label{table.term1d}

    \begin{tabular}{lllllll}

    \toprule

    Symbol& Harmonic Oscillator &Infinite Well&Illustrate \\

    \midrule

    $E_i$ &$\lambda$&$\lambda^2\pi^2\over 2$ & Energy \\

    $pdf(x)$ &$e^{-x^2}$ &1 & Sampling density compensation function\\

    \bottomrule

    \end{tabular}

    \end{table}

\subsubsection{1D harmonic oscillator}

The comparison of the 1D harmonic oscillator wavefunction and its error is shown in Figure.\ref{fig.psi.HSO.1d}, and the convergence process is shown in figure\ref{fig.Convergence.HSO.1d}, energy level and error analysis see Table.\ref{table.E.HSO.1d} (in $\hbar\omega_0$):

  \begin{figure}[H]
    \centering
    \subfigure[$E_0=0.5004$]{
      \label{fig:subfig:0}
      \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_0.pdf}}
    \hspace{-1em}
    \subfigure[$E_1=1.4998$]{
      \label{fig:subfig:1}
      \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_1.pdf}}
    \hspace{-1em}
    \subfigure[$E_2=2.4997$]{
        \label{fig:subfig:2}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_2.pdf}}
      \hspace{-1em}
      \vspace{-1.5em}

    \subfigure[$E_3=3.4989$]{
        \label{fig:subfig:3}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_3.pdf}}
        \hspace{-1em} %
    \subfigure[$E_4=4.4983$]{
        \label{fig:subfig:4}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_4.pdf}}
        \hspace{-1em}
    \subfigure[$E_5=5.4984$]{
        \label{fig:subfig:5}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_5.pdf}}
        \hspace{-1em}
        \vspace{-1.5em}

    \subfigure[$E_6=6.4986$]{
        \label{fig:subfig:6}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_6.pdf}}
        \hspace{-1em}
    \subfigure[$E_7=7.5006$]{
        \label{fig:subfig:7}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_7.pdf}}
        \hspace{-1em}
    \subfigure[$E_8=8.5021$]{
        \label{fig:subfig:8}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_8.pdf}}
        \hspace{-1em}
        \vspace{-1.5em}

    \subfigure[$E_3=9.5045$]{
        \label{fig:subfig:9}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_9.pdf}}
        \hspace{-1em}
    \subfigure[$E_4=10.4971$]{
        \label{fig:subfig:10}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_10.pdf}}
        \hspace{-1em}
    \subfigure[$E_5=11.5219$]{
        \label{fig:subfig:11}
        \includegraphics[scale=0.3]{picture/HSO_1d/psi_1d_11.pdf}}
        \hspace{-1em}
        \vspace{-1em}

    \subfigure[]{
    \label{fig_1HO:subfig:legend}
    \includegraphics[scale=0.8]{picture/legend_1d.pdf}}
    \caption{1D harmonic oscillator potential wave function. It can be seen that the maximum error of the wave function is within 2\%, and the larger errors occur in the places with the largest gradients, which is consistent with the literature \cite{F_Principle}.}

    \label{fig.psi.HSO.1d}
  \end{figure}

  \begin{figure}[H]
    \centering
    \subfigure[Energy Eigenvalue]{
      \label{fig:subfig:E}
      \includegraphics[scale=0.45]{picture/HSO_1d/eigen.pdf}}
    \hspace{-2em}
    \subfigure[Energy eigenvalue uncertainty]{
      \label{fig:subfig:DE}
      \includegraphics[scale=0.45]{picture/HSO_1d/DE.pdf}}
    \hspace{-2em}
    \subfigure[Total Loss Function]{
      \label{fig:subfig:L}
      \includegraphics[scale=0.45]{picture/HSO_1d/Loss.pdf}}
    \hspace{-2em}

    \subfigure[Residual of Equation]{
        \label{fig:subfig:F}
        \includegraphics[scale=0.45]{picture/HSO_1d/f_loss.pdf}}
      \hspace{-2em} %
    \subfigure[normalization condition]{
        \label{fig:subfig:N}
        \includegraphics[scale=0.45]{picture/HSO_1d/normal.pdf}}
        \hspace{-2em} %
    \subfigure[orthogonality condition]{
        \label{fig:subfig:O}
        \includegraphics[scale=0.45]{picture/HSO_1d/orth.pdf}}
        \hspace{-2em} %
    \caption{一维谐振子收敛曲线}
    \caption{Convergence curve of one-dimensional harmonic oscillator}
    \label{fig.Convergence.HSO.1d}
  \end{figure}

\begin{table}[H]\centering
  \caption{one-dimensional resonant potential energy level and error}
  \label{table.E.HSO.1d}
  \begin{tabular}{c c c c c c c}
    \toprule

    \makecell[c]{Energy \\Level} & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\
    \midrule
      0 &  0.5 &  0.5004 &  0.5004 &  0.0385 & 0.0004 &   0.0748 \\
      1 &  1.5 &  1.4998 &  1.4999 &  0.0231 & 0.0002 &  -0.0117 \\
      2 &  2.5 &  2.4997 &  2.4999 &  0.0230 & 0.0003 &  -0.0117 \\
      3 &  3.5 &  3.4989 &  3.4997 &  0.0394 & 0.0011 &  -0.0313 \\
      4 &  4.5 &  4.4983 &  4.4990 &  0.0453 & 0.0017 &  -0.0384 \\
      5 &  5.5 &  5.4984 &  5.4989 &  0.0503 & 0.0016 &  -0.0298 \\
      6 &  6.5 &  6.4986 &  6.4988 &  0.0530 & 0.0014 &  -0.0212 \\
      7 &  7.5 &  7.5006 &  7.4991 &  0.0888 & 0.0006 &   0.0076 \\
      8 &  8.5 &  8.5021 &  8.5004 &  0.1113 & 0.0021 &   0.0246 \\
      9 &  9.5 &  9.5045 &  9.5027 &  0.1432 & 0.0045 &   0.0469 \\
     10 & 10.5 & 10.4971 & 10.4919 &  0.1581 & 0.0029 &  -0.0274 \\
     11 & 11.5 & 11.5219 & 11.5155 &  0.2209 & 0.0219 &   0.1907 \\
    \bottomrule
    \end{tabular}
    \end{table}

\subsubsection{1D infinite deep potential well}

The analysis of the first 12 wave functions and errors of the 1D infinite deep potential well is shown in Figure.\ref{fig.psi.IW.1d}, and the convergence curve is shown in the Figure.\ref{fig.Convergence.IW.1d}, energy level see Table.\ref{table.E.IW.1d} (unit is $\hbar^2 \over {2\mu}$):

\begin{figure}[H]
  \centering
  \subfigure[$E_1=1.2394$]{
    \label{fig:subfig:S0}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_1.pdf}}
  \hspace{-1em}
  \subfigure[$E_2=4.9438$]{
    \label{fig:subfig:S1}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_2.pdf}}
  \hspace{-1em}
  \subfigure[$E_3=11.1026$]{
      \label{fig:subfig:S2}
      \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_3.pdf}}
    \hspace{-1em}
    \vspace{-1em}

  \subfigure[$E_4=19.7318$]{
      \label{fig:subfig:S3}
      \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_4.pdf}}
      \hspace{-1em}
  \subfigure[$E_5=30.8701$]{
      \label{fig:subfig:S4}
      \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_5.pdf}}
      \hspace{-1em}
  \subfigure[$E_6=44.4765$]{
      \label{fig:subfig:S5}
      \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_6.pdf}}
      \hspace{-1em}
      \vspace{-1em}

  \subfigure[$E_7=60.4013$]{
      \label{fig:subfig:S6}
      \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_7.pdf}}
      \hspace{-1em}
  \subfigure[$E_8=78.9671$]{
    \label{fig:subfig:S7}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_8.pdf}}
    \hspace{-1em}
  \subfigure[$E_9=99.9682$]{
    \label{fig:subfig:S8}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_9.pdf}}
    \hspace{-1em}
    \vspace{-1em}

  \subfigure[$E_10=123.2579$]{
    \label{fig:subfig:S9}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_10.pdf}
    }
    \hspace{-1em}
  \subfigure[$E_{11}=149.1855$]{
    \label{fig:subfig:S10}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_11.pdf}}
    \hspace{-1em}
  \subfigure[$E_{12}=177.5742$]{
    \label{fig:subfig:S11}
    \includegraphics[scale=0.3]{picture/IW_1d/psi_1d_12.pdf}}
    \hspace{-1em}
    \vspace{-1em}

  \subfigure[]{
    \label{fig:subfig:legend}
    \includegraphics[scale=0.8]{picture/legend_1d.pdf}}
  \caption{One-dimensional potential well wave function with infinite depth}
  \label{fig.psi.IW.1d}
\end{figure}

\begin{figure}[H]\centering
  \centering
  \subfigure[Energy Eigenvalue]{
    \label{fig:subfig:IE}
    \includegraphics[scale=0.45]{picture/IW_1d/eigen.pdf}}
  \hspace{-2em}
  \subfigure[Energy Eigenvalue uncertainty]{
    \label{fig:subfig:IDE}
    \includegraphics[scale=0.43]{picture/HSO_1d/DE.pdf}}
  \hspace{-2em}
  \subfigure[Total loss function]{
    \label{fig:subfig:IL}
    \includegraphics[scale=0.45]{picture/IW_1d/Loss.pdf}}
  \hspace{-2em}

  \subfigure[Residual of Equation]{
      \label{fig:subfig:IF}
      \includegraphics[scale=0.45]{picture/IW_1d/f_loss.pdf}}
    \hspace{-2em} %
  \subfigure[normalization condition]{
      \label{fig:subfig:IN}
      \includegraphics[scale=0.45]{picture/IW_1d/normal.pdf}}
      \hspace{-2em} %
  \subfigure[orthogonality condition]{
      \label{fig:subfig:IO}
      \includegraphics[scale=0.45]{picture/IW_1d/orth.pdf}}
      \hspace{-2em} %

    \caption{Convergence curve of one-dimensional infinite deep potential well}
  \label{fig.Convergence.IW.1d}
\end{figure}

\begin{table}\centering
  \caption{  Energy levels and errors of one-dimensional infinite deep potential well}
  \label{table.E.IW.1d}
  \begin{tabular}{c c c c c c c}
    \toprule

    \makecell[c]{Energy \\Level} & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\
    \midrule
      1 &   1.2337 &   1.2394 &   1.2335 &  0.0882 & 0.0057 &   0.4592 \\
      2 &   4.9348 &   4.9438 &   4.9349 &  0.0983 & 0.0090 &   0.1816 \\
      3 &  11.1033 &  11.1026 &  11.1036 &  0.1922 & 0.0007 &  -0.0064 \\
      4 &  19.7392 &  19.7318 &  19.7381 &  0.3127 & 0.0074 &  -0.0373 \\
      5 &  30.8425 &  30.8701 &  30.8425 &  0.4378 & 0.0276 &   0.0896 \\
      6 &  44.4132 &  44.4765 &  44.4131 &  0.5167 & 0.0633 &   0.1425 \\
      7 &  60.4513 &  60.4013 &  60.4471 &  0.9101 & 0.0500 &  -0.0828 \\
      8 &  78.9568 &  78.9671 &  78.9555 &  0.9797 & 0.0103 &   0.0130 \\
      9 &  99.9297 &  99.9682 &  99.9244 &  1.1430 & 0.0385 &   0.0385 \\
     10 & 123.3701 & 123.2579 & 123.3466 &  1.8754 & 0.1122 &  -0.0909 \\
     11 & 149.2778 & 149.1855 & 149.2585 &  1.9479 & 0.0923 &  -0.0618 \\
     12 & 177.6529 & 177.5742 & 177.7147 &  3.5324 & 0.0787 &  -0.0443 \\
    \bottomrule
    \end{tabular}
  \end{table}

\subsection{Degeneracy Problem}

For degeneracy problems, we mainly consider the possibility of solving multiple eigenstates and eigenvalues with degeneracy at the same time. The 2D circular square infinite deep potential is representative of the finite margin problem, and the 3D harmonic oscillator is the representative of the infinite margin problem.

The both problems are analyzed as following Table\ref{table.problem2d}.

\begin{table}[H]\centering \caption{2D problem}\label{table.problem2d}

  \begin{tabular}{lllllll}

  \toprule

  Symbol& 3D Harmonic Oscillator &2D Infinite Well&Illustrate \\

  \midrule

  $V(x)$ &${1\over 2} r^2$ & $0,r\in{other}; \infty,r\in{(0,1)}$&Potential\\

  $\mathcal{B} c$ &$\Psi(\vec{r})\stackrel{r\rightarrow \pm\infty}{\longrightarrow}0$ &$\Psi(\vec{r})\stackrel{r=1}{\longrightarrow}0$&Boundary Condition\\

  $\mathcal{B}(x)$ &$e^{-{1\over2}r^2}$ & $(1-r^2)$&Boundary Term\\

  $\Psi^R_n$ &$N_nH_n(x_1)H_n(x_2)H_n(x_3)e^{-{r^2\over2}}$ & $N_nJ_{|m|}(x_{nm}r)sin(m\phi+\phi_0)$&Real Wavefunction\\

  $E^R_n$ &$n+1$ & ${{x_{nm}^2}\over2}$&Real Energy\\

  \bottomrule

  \end{tabular}

  \end{table}

In the table above, $H_n(x)$ is the Ermi polynomial, $J_{|m|} (x_{nm}r)$ is the Besell function, $x_{nm}$ is the 0-point position of the first non-origin of the Bezier function. $\phi_0$ is the initial phase, which can orthogonalize the two states of degeneracy.

The neural network solution and loss function for the degeneracy problems are similar to those for the 1D problem. In the above equation, in order to facilitate convergence, it is necessary to redesign the sampling method and the eigenvalue form design for the problem, as shown in the following Table\ref{table.term2d}.

  \begin{table}[H]\centering \caption{Key indicators of one-dimensional potential training}
    \label{table.term2d}

    \begin{tabular}{lllllll}

    \toprule

    Symbol& Harmonic Oscillator &Infinite Well&Illustrate \\

    \midrule

    $E_n$ &$\lambda$&$\lambda^2$ & Energy \\

    $pdf(x)$ &$e^{-r^2}$ &1 & Sampling density compensation function\\

    \bottomrule

    \end{tabular}

    \end{table}

The 2D circular infinite potential well problem is similar to the cylindrical waveguide problem that the solution function corresponds to a wave mode, the energy level corresponds to the wavelength, and the ground state energy corresponds to the cutoff frequency.

\subsubsection{3D HSO}

The first 20 wave functions of the 3D harmonic oscillator potential are shown in the Figure.[\ref{fig.psi.HSO.3dA1},\ref{fig.psi.HSO.3dA2},\ref{fig.psi.HSO.3dA3},\ref{fig.psi.HSO.3dA4}], and the convergence curve is shown in the figure\ref{fig.Convergence.HSO.3d},Energy levels are shown in table\ref{table.E.HSO.3d} (unit is $\hbar^2 over {2\mu}$):

\begin{figure}[H]\centering
\centering
\subfigure[Energy Eigenvalue]{
  \label{fig_3H:subfig:IE}
  \includegraphics[scale=0.45]{picture/HSO_3d/eigen.pdf}}
\hspace{-3em}
\subfigure[Energy eigenvalue uncertainty]{
  \label{fig_3H:subfig:IDE}
  \includegraphics[scale=0.45]{picture/HSO_3d/DE.pdf}}
\hspace{-2em}
\subfigure[Total loss function]{
  \label{fig_3H:subfig:IL}
  \includegraphics[scale=0.45]{picture/HSO_3d/Loss.pdf}}
\hspace{-2em}

\subfigure[Residual of Equation]{
    \label{fig_3H:subfig:IF}
    \includegraphics[scale=0.45]{picture/HSO_3d/f_loss.pdf}}
  \hspace{-2em} %
\subfigure[normalization condition]{
    \label{fig_3H:subfig:IN}
    \includegraphics[scale=0.45]{picture/HSO_3d/normal.pdf}}
    \hspace{-2em} %
\subfigure[orthogonality condition]{
    \label{fig_3H:subfig:IO}
    \includegraphics[scale=0.45]{picture/HSO_3d/orth.pdf}}
    \hspace{-2em} %
\caption{Convergence curve of three dimensional harmonic oscillator potential}
\label{fig.Convergence.HSO.3d}
\end{figure}

\begin{table}\centering
  \caption{Potential energy level and error of three-dimensional harmonic oscillator}
  \label{table.E.HSO.3d}
  \begin{tabular}{rrrrrrrrrr}
    \toprule

     \makecell[c]{Energy \\Level} &  n1 &  n2 &  n3 & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\

    \midrule
     0 &   0 &   0 &   0 &  1.5 & 1.4983 & 1.5004 &  0.0797 & 0.0017 &  -0.1104 \\
     1 &   1 &   0 &   0 &  2.5 & 2.5003 & 2.5064 &  0.1499 & 0.0003 &   0.0102 \\
     1 &   0 &   1 &   0 &  2.5 & 2.5035 & 2.5053 &  0.1470 & 0.0035 &   0.1413 \\
     1 &   0 &   0 &   1 &  2.5 & 2.5043 & 2.5076 &  0.1530 & 0.0043 &   0.1725 \\
     2 &   1 &   1 &   0 &  3.5 & 3.4961 & 3.5103 &  0.1892 & 0.0039 &  -0.1104 \\
     2 &   1 &   0 &   1 &  3.5 & 3.4971 & 3.5065 &  0.1849 & 0.0029 &  -0.0839 \\
     2 &   0 &   1 &   1 &  3.5 & 3.4993 & 3.5096 &  0.1899 & 0.0007 &  -0.0207 \\
     2 &   0 &   0 &   0 &  3.5 & 3.5009 & 3.5078 &  0.1940 & 0.0009 &   0.0269 \\
     2 &   0 &   0 &   0 &  3.5 & 3.5014 & 3.5076 &  0.1894 & 0.0014 &   0.0392 \\
     2 &   0 &   0 &   0 &  3.5 & 3.5025 & 3.5094 &  0.1846 & 0.0025 &   0.0716 \\
     3 &   0 &   0 &   0 &  4.5 & 4.4976 & 4.5109 &  0.2163 & 0.0024 &  -0.0536 \\
     3 &   0 &   0 &   0 &  4.5 & 4.4980 & 4.5111 &  0.2266 & 0.0020 &  -0.0450 \\
     3 &   0 &   0 &   0 &  4.5 & 4.4987 & 4.5140 &  0.2293 & 0.0013 &  -0.0282 \\
     3 &   0 &   0 &   0 &  4.5 & 4.5000 & 4.5098 &  0.2109 & 0.0000 &  -0.0000 \\
     3 &   0 &   0 &   0 &  4.5 & 4.5001 & 4.5142 &  0.2221 & 0.0001 &   0.0012 \\
     3 &   0 &   0 &   0 &  4.5 & 4.5009 & 4.5153 &  0.2183 & 0.0009 &   0.0199 \\
     3 &   0 &   0 &   0 &  4.5 & 4.5022 & 4.5108 &  0.2194 & 0.0022 &   0.0480 \\
     3 &   0 &   0 &   0 &  4.5 & 4.5037 & 4.5153 &  0.2172 & 0.0037 &   0.0818 \\
     4 &   0 &   0 &   0 &  5.5 & 5.5009 & 5.5123 &  0.2436 & 0.0009 &   0.0164 \\
     4 &   0 &   0 &   0 &  5.5 & 5.5015 & 5.5022 &  0.2729 & 0.0015 &   0.0274 \\
    \bottomrule
    \end{tabular}
  \end{table}

\begin{figure}[H]
    \centering
    \vspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S0a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_0_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S0b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_0_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S0c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_0_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S1a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_1_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S1b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_1_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S1c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_1_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S2a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_2_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S2b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_2_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S2c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_2_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S3a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_3_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S3b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_3_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S3c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_3_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S4a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_4_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S4b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_4_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S4c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_4_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em} %

    \subfigure{
      \label{fig_3HSO:subfig:S5a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_5_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S5b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_5_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S5c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_5_C1.pdf}}
    \hspace{-2em}
    \caption{3-dimensional HSO potential well wave function}
    \label{fig.psi.HSO.3dA1}
  \end{figure}

\begin{figure}[H]
    \centering
    \vspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S6a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_6_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S6b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_6_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S6c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_6_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S7a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_7_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S7b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_7_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S7c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_7_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S8a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_8_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S8b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_8_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S8c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_8_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S9a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_9_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S9b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_9_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S9c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_9_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S10a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_10_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S10b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_10_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S10c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_10_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em} %

    \subfigure{
      \label{fig_3HSO:subfig:S11a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_11_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S11b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_11_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S11c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_11_C1.pdf}}
    \hspace{-2em}
    \caption{3-dimensional HSO potential well wave function}
    \label{fig.psi.HSO.3dA2}
  \end{figure}

\begin{figure}[H]
    \centering
    \vspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S12a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_12_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S12b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_12_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S12c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_12_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S13a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_13_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S13b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_13_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S13c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_13_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S14a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_14_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S14b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_14_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S14c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_14_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S15a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_15_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S15b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_15_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S15c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_15_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S16a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_16_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S16b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_16_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S16c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_16_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em} %

    \subfigure{
      \label{fig_3HSO:subfig:S17a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_17_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S17b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_17_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S17c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_17_C1.pdf}}
    \hspace{-2em}
    \caption{3-dimensional HSO potential well wave function}
    \label{fig.psi.HSO.3dA3}
  \end{figure}

\begin{figure}[H]
    \centering
    \vspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S18a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_18_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S18b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_18_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S18c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_18_C1.pdf}}
    \hspace{-2em}
    \vspace{-2em}

    \subfigure{
      \label{fig_3HSO:subfig:S19a}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_19_A1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S19b}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_19_B1.pdf}}
    \hspace{-2em}
    \subfigure{
      \label{fig_3HSO:subfig:S19c}
      \includegraphics[scale=0.3]{picture/HSO_3d/psi_3d_19_C1.pdf}}
    \hspace{-2em}
    \caption{3-dimensional HSO potential well wave function}
    \label{fig.psi.HSO.3dA4}
  \end{figure}

\subsubsection{2D IW}

The first 12 wave functions of the 2D circular square infinite deep potential well are shown in the Figure.[\ref{fig.psi.IW.2dA1},\ref{fig.psi.IW.2dA2}], and the convergence curve is shown in the figure\ref{fig.Convergence.IW.2d},Energy levels are shown in table\ref{table.E.IW.2d} (unit is $\hbar^2 over {2\mu}$):

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S1}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_0_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S01}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_0_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S02}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_0_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S10}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_1_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S11}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_1_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S12}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_1_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S20}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_2_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S21}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_2_B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S22}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_2_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S30}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_3_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S31}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_3_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S32}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_3_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S40}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_4_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S41}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_4_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S42}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_4_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S50}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_5_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S51}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_5_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S52}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_5_C.pdf}}
  \hspace{-2em}
  \caption{Two-dimensional infinite deep potential well wave function}
  \label{fig.psi.IW.2dA1}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S6a}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_6_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S6b}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_6_B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S6c}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_6_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S7a}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_7_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S7b}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_7_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S7c}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_7_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S8a}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_8_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S8b}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_8_B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S8c}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_8_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S9a}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_9_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S9b}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_9_B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S9c}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_9_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S10a}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_10_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S10b}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_10_B.pdf}}
  \hspace{-2em}
  \subfigure{
      \label{fig_2I:subfig:S10c}
      \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_10_C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2I:subfig:S11a}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_11_A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S11b}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_11_B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2I:subfig:S11c}
    \includegraphics[scale=0.3]{picture/IW_2d/psi_2d_11_C.pdf}}
  \hspace{-2em}
  \caption{Two-dimensional infinite deep potential well wave function}
  \label{fig.psi.IW.2dA2}
\end{figure}

\begin{figure}[H]
  \centering
  \subfigure[Energy Eigenvalue]{
    \label{fig_2I:subfig:IE}
    \includegraphics[scale=0.45]{picture/IW_2d/eigen.pdf}}
  \hspace{-2em}
  \subfigure[Energy eigenvalue uncertainty]{
    \label{fig_2I:subfig:IDE}
    \includegraphics[scale=0.45]{picture/IW_2d/DE.pdf}}
  \hspace{-2em}
  \subfigure[Total loss function]{
    \label{fig_2I:subfig:IL}
    \includegraphics[scale=0.45]{picture/IW_2d/Loss.pdf}}
  \hspace{-2em}

  \subfigure[Residual of Equation]{
      \label{fig_2I:subfig:IF}
      \includegraphics[scale=0.45]{picture/IW_2d/f_loss.pdf}}
    \hspace{-2em} %
  \subfigure[normalization condition]{
      \label{fig_2I:subfig:IN}
      \includegraphics[scale=0.45]{picture/IW_2d/normal.pdf}}
      \hspace{-2em} %
  \subfigure[orthogonality condition]{
      \label{fig_2I:subfig:IO}
      \includegraphics[scale=0.45]{picture/IW_2d/orth.pdf}}
      \hspace{-2em} %
  \caption{Convergence curve of 2D infinite deep potential well}
  \label{fig.Convergence.IW.2d}
\end{figure}

\begin{table}\centering
  \caption{Energy levels and errors of 2D infinite deep potential well.The first 12 states contain $E_1, E_6$ two non-degenerate states have $m=0$, and the other states are double degeneracy $(E_2, E_3)$, $(E_4, E_5)$, $(E_7, E_8)$, $(E_9, E_{10})$, $(E_{11}, E_{12})$, which is completely consistent with the theoretical results.}

  \label{table.E.IW.2d}
  \begin{tabular}{rrrrrrr}
    \toprule

    \makecell[c]{Energy \\Level} & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\
    \midrule
      1 &  2.8916 &  2.8968 &  2.8916 &  0.0284 & 0.0052 &   0.1792 \\
      2 &  7.3410 &  7.3405 &  7.3405 &  0.0609 & 0.0004 &  -0.0059 \\
      3 &  7.3410 &  7.3428 &  7.3407 &  0.0475 & 0.0018 &   0.0247 \\
      4 & 13.1873 & 13.1839 & 13.1870 &  0.0929 & 0.0034 &  -0.0261 \\
      5 & 13.1873 & 13.1873 & 13.1866 &  0.1153 & 0.0000 &  -0.0001 \\
      6 & 15.2356 & 15.2373 & 15.2349 &  0.1423 & 0.0017 &   0.0110 \\
      7 & 20.3532 & 20.3472 & 20.3507 &  0.2409 & 0.0060 &  -0.0294 \\
      8 & 20.3532 & 20.3483 & 20.3523 &  0.1923 & 0.0050 &  -0.0244 \\
      9 & 24.6092 & 24.6091 & 24.6052 &  0.3359 & 0.0001 &  -0.0005 \\
     10 & 24.6092 & 24.6103 & 24.6036 &  0.3984 & 0.0011 &   0.0045 \\
     11 & 28.7915 & 28.7609 & 28.7830 &  0.5298 & 0.0305 &  -0.1060 \\
     12 & 28.7915 & 28.7944 & 28.7840 &  0.5455 & 0.0029 &   0.0100 \\
    \bottomrule
    \end{tabular}
  \end{table}

\subsection{2D HSO}

\begin{figure}[H]
  \centering
  \subfigure[Energy eigenvalue uncertainty]{
    \label{fig_2s:subfig:DE}
    \includegraphics[scale=0.7]{picture/HSO_2d/DE.pdf}}
  \hspace{-2em}
  \subfigure[Residual of Equation]{
      \label{fig_2s:subfig:F}
      \includegraphics[scale=0.6]{picture/HSO_1d/f_loss.pdf}}
    \hspace{-2em} %

  \subfigure[normalization condition]{
      \label{fig_2s:subfig:N}
      \includegraphics[scale=0.6]{picture/HSO_1d/normal.pdf}}
      \hspace{-2em} %
  \subfigure[orthogonality condition]{
      \label{fig_2s:subfig:O}
      \includegraphics[scale=0.6]{picture/HSO_1d/orth.pdf}}
      \hspace{-2em} %
  \caption{Convergence curve of two dimensional harmonic oscillator}
  \label{fig.Convergence.HSO.2dA}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S0a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_0A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S0b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_0A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S0c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_0C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S1a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_1A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S1b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_1B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S1c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_1C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S2a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_2A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S2b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_2B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S2c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_2C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S3a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_3A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S3b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_3B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S3c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_3C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S4a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_4A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S4b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_4B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S4c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_4C.pdf}}
  \hspace{-2em}
  \vspace{-2em} %

  \subfigure{
    \label{fig_2HSO:subfig:S5a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_5A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S5b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_5B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S5c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_5C.pdf}}
  \hspace{-2em}
  \caption{Two-dimensional HSO potential well wave function}
  \label{fig.psi.HSO.2dA1}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S6d}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_6A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S6b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_6B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S6c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_6C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S7a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_7A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S7b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_7A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S7c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_7C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S8a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_8A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S8b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_8B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S8c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_8C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S9a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_9A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S9b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_9B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S9c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_9C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S10a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_10A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S10b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_10B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S10c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_10C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S11a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_11A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S11b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_11B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S11c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_11C.pdf}}
  \hspace{-2em}
  \caption{Two-dimensional HSO potential well wave function}
  \label{fig.psi.HSO.2dA2}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S12d}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_12A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S12b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_12B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S12c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_12C.pdf}}
  \hspace{-2em}
  \vspace{-2em} %

  \subfigure{
    \label{fig_2HSO:subfig:S13a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_13A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S13b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_13A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S13c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_13C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S14a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_14A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S14b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_14B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S14c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_14C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S15a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_15A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S15b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_15B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S15c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_15C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S16a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_16A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S16b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_16B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S16c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_16C.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_2HSO:subfig:S17a}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_17A.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S17b}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_17B.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_2HSO:subfig:S17c}
    \includegraphics[scale=0.3]{picture/HSO_2d/psi_2d_17C.pdf}}
  \hspace{-2em}
  \vspace{-2em}
  \caption{Two-dimensional HSO potential well wave function}
  \label{fig.psi.HSO.2dA3}
\end{figure}

\subsection{3D IW}
\begin{figure}[H]\centering
  \centering
  \subfigure[Energy eigenvalue uncertainty]{
    \label{fig_3I:subfig:IDE}
    \includegraphics[scale=0.6]{picture/IW_3d/DE.pdf}}
  \hspace{-2em}
  \subfigure[Residual of Equation]{
      \label{fig_3I:subfig:IF}
      \includegraphics[scale=0.6]{picture/IW_3d/f_loss.pdf}}
    \hspace{-2em} %

  \subfigure[normalization condition]{
      \label{fig_3I:subfig:IN}
      \includegraphics[scale=0.6]{picture/IW_3d/normal.pdf}}
      \hspace{-2em} %
  \subfigure[orthogonality condition]{
      \label{fig_3I:subfig:IO}
      \includegraphics[scale=0.6]{picture/IW_3d/orth.pdf}}
      \hspace{-2em} %
  \caption{Convergence curve of three-dimensional infinite deep potential well}
  \label{fig.Convergence.IW.3dA}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S0a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_0_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S0b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_0_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S0c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_0_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S1a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_1_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S1b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_1_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S1c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_1_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S2a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_2_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S2b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_2_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S2c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_2_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S3a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_3_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S3b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_3_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S3c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_3_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S4a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_4_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S4b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_4_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S4c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_4_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em} %

  \subfigure{
    \label{fig_3IW:subfig:S5a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_5_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S5b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_5_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S5c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_5_C1.pdf}}
  \hspace{-2em}
  \caption{3-dimensional HSO potential well wave function}
  \label{fig.psi.IW.3dA1}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S6a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_6_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S6b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_6_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S6c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_6_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S7a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_7_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S7b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_7_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S7c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_7_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S8a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_8_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S8b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_8_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S8c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_8_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S9a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_9_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S9b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_9_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S9c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_9_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3IW:subfig:S10a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_10_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S10b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_10_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S10c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_10_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em} %

  \subfigure{
    \label{fig_3IW:subfig:S11a}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_11_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S11b}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_11_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3IW:subfig:S11c}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_11_C1.pdf}}
  \hspace{-2em}
  \caption{3-dimensional HSO potential well wave function}
  \label{fig.psi.IW.3dA2}
\end{figure}

\subsection{3D H}
\begin{figure}[H]\centering
  \centering
  \subfigure[Energy eigenvalue uncertainty]{
    \label{fig_3Hy:subfig:IDE}
    \includegraphics[scale=0.6]{picture/H_3d/DE.pdf}}
  \hspace{-2em}
  \subfigure[Residual of Equation]{
      \label{fig_3Hy:subfig:IF}
      \includegraphics[scale=0.6]{picture/H_3d/f_loss.pdf}}
    \hspace{-2em} %

  \subfigure[normalization condition]{
      \label{fig_3Hy:subfig:IN}
      \includegraphics[scale=0.6]{picture/H_3d/normal.pdf}}
      \hspace{-2em} %
  \subfigure[orthogonality condition]{
      \label{fig_3Hy:subfig:IO}
      \includegraphics[scale=0.6]{picture/H_3d/orth.pdf}}
      \hspace{-2em} %
  \caption{Hydrogen potential convergence curve}
  \label{fig.Convergence.H.3dA}
\end{figure}

\begin{figure}[H]
  \centering
  \vspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S0a}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_0_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S0b}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_0_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S0c}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_0_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3H:subfig:S1a}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_1_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S1b}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_1_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S1c}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_1_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3H:subfig:S2a}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_2_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S2b}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_2_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S2c}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_2_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3H:subfig:S3a}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_3_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S3b}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_3_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S3c}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_3_C1.pdf}}
  \hspace{-2em}
  \vspace{-2em}

  \subfigure{
    \label{fig_3H:subfig:S4a}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_4_A1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S4b}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_4_B1.pdf}}
  \hspace{-2em}
  \subfigure{
    \label{fig_3H:subfig:S4c}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_4_C1.pdf}}
  \hspace{-2em}
  \caption{3-dimensional HSO potential well wave function}
  \label{fig.psi.H.3dA1}
\end{figure}
\section{Conclusion}

An approach based on neural networks to solving the static Schr\"{o}dinger equation is presented in this paper. Its basic principle and algorithm flow are given. And seven numerical experiments which include 1D-3D harmonic oscillators and infinite potential wells, and the hydrogen-like potential are taken as examples to describe the specific method and related characteristics. For each example, the specific solution process algorithm differs only in boundary conditions, sampling methods, and potential function settings, and different network scales and learning rates should be set for different problems. Comparing the convergence process and error analysis of the harmonic oscillator with the literature \cite{liNeuralnetworkbasedMultistateSolver2021}, it can be seen that the convergence speed and the final energy accuracy have improved, which can explain the superiority of this algorithm. The 2D circular square infinite potential well is equivalent to the cylindrical waveguide problem in the electromagnetic field technology, and the three-dimensional infinite spherical square potential well and the spherical resonator are equivalent problems, which shows the engineering practicability of the algorithm. The solution of the hydrogen-like atom problem provides a new basis for the existing atomic and molecular structure calculation.

The energy level convergence process diagram of the seven groups of experiments Figure.\{\ref{fig_2s:subfig:E}, \ref{fig_3I:subfig:IE} , \ref{fig_3Hy:subfig:IE}\}, the overall convergence process is from low energy level to high energy level convergence. Considering the premise of the frequency assumption of the wave function, this is consistent with the frequency principle proposed by the literature \cite{F_Principle}. Of course, it also shows the correctness of the wave function frequency assumption. However, because the neural network structure used in this article is the simplest fully connected neural network, when one of the energy levels changes greatly, because all the energy levels essentially share most of the network parameters, the trained energy levels will also change slightly, and may even converge to other states. So the following work can be solved by designing a targeted neural network structure for this.
\section{Results and discussion}

A few examples are given to demonstrate the algorithm's effectiveness. It includes the problem of the infinite deep potential well, the harmonic oscillator problem in 1D, 2D, 3D, and the problem of the three-dimensional hydrogen atom. All problems are solved by using Cartesian coordinates. According to the analysis of the second section, for any potential field, it is necessary to determine the potential function form and boundary conditions, and then set reasonable boundary conditions based on those parameters. In order to speed up convergence, it is important to design reasonable eigenvalue terms according to the eigenvalue's possible form. In addition, it is important to select an appropriate sampling scheme based on the characteristics of the potential field. These steps are also illustrated in the examples below.

The training process of the algorithm as a whole is divided into two steps. The first step is to use a large coefficient of orthogonal normalization conditions and equation residuals with a small number of sampling points for each output can converge quickly to each other orthogonal energy levels. The second is to improve the final convergence accuracy by reducing these three coefficients, increasing coefficient of the eigenvalue uncertainty and increasing sampling points.

In all seven cases, a single training was used to obtain multiple states except for the 2D harmonic oscillator.The 2D harmonic oscillator obtained all 18 states by three trainings after using the single training to obtain six states each time. The following are some numerical experiments, and the other four groups of experiments are shown in the appendix. At the same time, more convergence process data and wave function error comparison are also in the appendix.

\subsection{Two dimensional harmonic oscillator problem}

Under atomic units, the 2D harmonic oscillator potential has the following form.

\begin{equation}\label
    & V(x)={1\over 2} r^2
    \end{equation}

where $r^2=\sum_i^2{x_i^2}$. The boundary condition term attached to the network is $\mathcal{B}(x)=e^{-{1\over2}r^2}$.

The real wave functions and energy levels are:

$$\Psi^R_n=N_nH_n(x_1)H_n(x_2)e^{-{r^2\over2}},E^R_n=n+1$$

where $H_n(x)$ is confluent Ermi polynomial, and $N_{n}$ is normalization coefficient.

In this experiment, we used a five-layer fully connected neural network, with 150 nodes in each layer of the middle layer and 6 nodes in the output layer. The total training was three times, and six states were obtained each time. In the second training, the six states obtained in the first training should be added to the loss function as the orthogonal condition of the known wave function and the proposed wave function. And then the third time you have to add the first two states. In order to facilitate convergence, it is necessary to re-design sampling methods for the problem, including $pdf(x)= e ^{-x^2}$and eigenvalue form design $E = \lambda$.

The 2D harmonic oscillator wave function is shown in the Figure.\ref{fig.psi.HSO.2d}, and the convergence curve is shown in the Figure.\ref{fig_2s:subfig:E}, Energy levels are shown in Table.\ref{table.E.HSO.2d} (unit $\hbar\omega_0$).

\begin{figure}[ht]
  \centering
  \subfigure[$E_0=1.0001$]{
    \label{fig_2s:subfig:00}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_0A.pdf}}
  \hspace{-3em}
  \subfigure[$E_1=2.0000$]{
    \label{fig_2s:subfig:01}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_1A.pdf}}
  \hspace{-3em}
  \subfigure[$E_2=2.0003$]{
      \label{fig_2s:subfig:02}
      \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_2A.pdf}}
    \hspace{-3em}
  \subfigure[$E_0=2.9997$]{
    \label{fig_2s:subfig:10}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_3A.pdf}}
  \hspace{-3em}
  \vspace{-2em}

  \subfigure[$E_1=2.9999$]{
    \label{fig_2s:subfig:11}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_4A.pdf}}
  \hspace{-3em}
  \subfigure[$E_2=3.0004$]{
      \label{fig_2s:subfig:12}
      \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_5A.pdf}}
  \hspace{-3em}
  \subfigure[$E_0=4.0000$]{
    \label{fig_2s:subfig:20}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_6A.pdf}}
  \hspace{-3em}
  \subfigure[$E_1=4.0003$]{
    \label{fig_2s:subfig:21}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_7A.pdf}}
  \hspace{-3em}
  \vspace{-2em}

  \subfigure[$E_2=4.0009$]{
      \label{fig_2s:subfig:22}
      \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_8A.pdf}}
  \hspace{-3em}
  \subfigure[$E_0=4.9998$]{
    \label{fig_2s:subfig:30}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_9A.pdf}}
  \hspace{-3em}
  \subfigure[$E_1=5.0010$]{
    \label{fig_2s:subfig:31}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_10A.pdf}}
  \hspace{-3em}
  \subfigure[$E_2=5.0011$]{
      \label{fig_2s:subfig:32}
      \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_11A.pdf}}
  \hspace{-3em}
  \vspace{-2em}

  \subfigure[$E_2=6.0001$]{
    \label{fig_2s:subfig:13}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_12A.pdf}}
  \hspace{-3em} %
  \subfigure[$E_0=6.0007$]{
    \label{fig_2s:subfig:14}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_13A.pdf}}
  \hspace{-3em}
  \subfigure[$E_1=6.0036$]{
    \label{fig_2s:subfig:15}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_14A.pdf}}
  \hspace{-3em}
  \subfigure[$E_2=7.9999$]{
      \label{fig_2s:subfig:16}
      \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_15A.pdf}}
  \hspace{-3em} %
  \vspace{-2em} %

  \subfigure[$E_2=8.0010$]{
    \label{fig_2s:subfig:17}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_16A.pdf}}
  \hspace{-3em} %
  \subfigure[$E_0=9.0007$]{
    \label{fig_2s:subfig:18}
    \includegraphics[scale=0.27]{picture/HSO_2d/psi_2d_17A.pdf}}
  \hspace{-3em}

  \subfigure[Energy Eigenvalue]{
    \label{fig_2s:subfig:E}
    \includegraphics[scale=0.6]{picture/HSO_2d/eigen.pdf}}
  \hspace{-2em}
  \subfigure[Total loss function]{
    \label{fig_2s:subfig:L}
    \includegraphics[scale=0.6]{picture/HSO_2d/Loss.pdf}}

  \caption{2D harmonic oscillator potential wave function.The x-coordinate is X, and the range is minus [10,10]. Y, and the range is also negative [10,10].}

  \label{fig.psi.HSO.2d}
\end{figure}

\begin{table}[ht]\centering
  \caption{Potential energy level and error of 2D harmonic oscillator}
  \label{table.E.HSO.2d}
  \begin{tabular}{rrrrrrrrr}
    \toprule
    \makecell[c]{Energy \\Level} & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\
    \midrule
     0 &  1.0 & 1.0001 & 0.4670 &  0.4673 & 0.0001 &   0.0103 \\
     1 &  2.0 & 2.0000 & 0.9353 &  0.6612 & 0.0000 &  -0.0003 \\
     1 &  2.0 & 2.0003 & 0.9354 &  0.6626 & 0.0003 &   0.0142 \\
     2 &  3.0 & 2.9997 & 1.3813 &  0.9459 & 0.0003 &  -0.0097 \\
     2 &  3.0 & 2.9999 & 1.3760 &  0.9442 & 0.0001 &  -0.0042 \\
     2 &  3.0 & 3.0004 & 1.4024 &  0.8163 & 0.0004 &   0.0135 \\
     3 &  4.0 & 4.0000 & 1.8367 &  1.0595 & 0.0000 &   0.0009 \\
     3 &  4.0 & 4.0003 & 1.7640 &  1.2673 & 0.0003 &   0.0083 \\
     3 &  4.0 & 4.0009 & 1.7825 &  1.2616 & 0.0009 &   0.0223 \\
     4 &  5.0 & 4.9998 & 2.1336 &  1.5985 & 0.0002 &  -0.0034 \\
     4 &  5.0 & 5.0010 & 2.2739 &  1.2586 & 0.0010 &   0.0191 \\
     4 &  5.0 & 5.0011 & 2.1617 &  1.5955 & 0.0011 &   0.0225 \\
     5 &  6.0 & 6.0001 & 2.5043 &  1.8611 & 0.0001 &   0.0010 \\
     5 &  6.0 & 6.0007 & 2.4445 &  1.9202 & 0.0007 &   0.0122 \\
     5 &  6.0 & 6.0036 & 2.4665 &  1.9220 & 0.0036 &   0.0604 \\
     7 &  8.0 & 7.9999 & 3.4346 &  2.0101 & 0.0001 &  -0.0017 \\
     7 &  8.0 & 8.0010 & 3.3732 &  2.1200 & 0.0010 &   0.0126 \\
     8 &  9.0 & 9.0007 & 3.8576 &  2.1540 & 0.0007 &   0.0073 \\
    \bottomrule
    \end{tabular}
    \end{table}

\subsection{three-dimensional round square infinite potential well problem}

Under atomic units, the 3D round square infinite potential well potential has the following form.

$$V(r)=
\begin{cases}
\infty,\quad r>1\\
0,\quad r<1
\end{cases}$$

where $r=\sqrt{\sum_{i=1}^3{x_i^2}}$. The boundary condition term attached to the network is $\mathcal{B}(x)=(1-r^2)$.

The real wave functions and energy levels are:

$$\Psi^R_n=N_nJ_{l}(x_{nl}r)Y_{lm}(\theta,\phi)$$

where $J_{l}(x_{nl}r)$ is the spherical Besell function, $x_{nl}$ is the 0-point position of the first non-origin of the Bezier function, and $N_{n}$ is normalization coefficient.

In this experiment, we use a five-layer fully connected neural network, with 120 nodes in each layer of the middle layer and 12 nodes in the output layer. Twelve states can be obtained in a single training. In order to facilitate convergence, it is necessary to re-design sampling methods for the problem, including $pdf (x)=1$, and eigenvalue form design $E = \lambda^2$.

The 3D sphere infinite potential well problem is similar to the sphere resonant cavity problem, the solution function corresponds to wave mode, the energy level corresponds to the wavelength, and the ground state energy corresponds to the cutoff frequency.

The first 12 wave functions Figure.\ref{fig.psi.IW.3d}, the convergence curve is shown in the Figure.\ref{fig_3I:subfig:IE},Energy levels are shown in Table.\ref{table.E.IW.3d} (unit is $\hbar^2 \over {2\mu}$):

\begin{figure}[ht]
  \centering
  \subfigure[$E_1=4.9348$]{
    \label{fig_3I:subfig:S1}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_0_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_2=10.0952$]{
    \label{fig_3I:subfig:S2}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_1_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_2=10.0954$]{
    \label{fig_3I:subfig:S3}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_2_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_2=10.0953$]{
    \label{fig_3I:subfig:S4}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_3_A1.pdf}}
  \hspace{-2em}

  \subfigure[$E_3=16.6077$]{
    \label{fig_3I:subfig:S5}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_4_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_3=16.6092$]{
      \label{fig_3I:subfig:S6}
      \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_5_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_3=16.6086 $]{
    \label{fig_3I:subfig:S7}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_6_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_4=16.6084$]{
    \label{fig_3I:subfig:S8}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_7_A1.pdf}}
  \hspace{-2em}

  \subfigure[$E_4=16.6094$]{
      \label{fig_3I:subfig:S9}
      \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_8_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_5=19.7381$]{
    \label{fig_3I:subfig:S10}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_9_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_6=24.4164$]{
    \label{fig_3I:subfig:S11}
    \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_10_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_6=24.4136$]{
      \label{fig_3I:subfig:S12}
      \includegraphics[scale=0.3]{picture/IW_3d/psi_3d_11_A1.pdf}}
  \hspace{-2em}

  \subfigure[Energy Eigenvalue]{
    \label{fig_3I:subfig:IE}
    \includegraphics[scale=0.6]{picture/IW_3d/eigen.pdf}}
  \hspace{-2em}
  \subfigure[Total loss function]{
    \label{fig_3I:subfig:IL}
    \includegraphics[scale=0.6]{picture/IW_3d/Loss.pdf}}
  \hspace{-2em}
  \caption{Three dimensional infinite deep potential well wave function}
  \label{fig.psi.IW.3d}
\end{figure}

\begin{table}\centering
  \caption{Three dimensional infinite deep potential well energy levels and errors}
  \label{table.E.IW.3d}
  \begin{tabular}{rrrrrrr}
    \toprule

    \makecell[c]{Energy \\Level} & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\

    \midrule
      1 &  4.9348 &  4.9298 &  4.9348 &  0.0257 & 0.0050 &  -0.1021 \\
      2 & 10.0935 & 10.0849 & 10.0952 &  0.0606 & 0.0086 &  -0.0854 \\
      2 & 10.0935 & 10.0912 & 10.0954 &  0.0957 & 0.0023 &  -0.0227 \\
      2 & 10.0935 & 10.1021 & 10.0953 &  0.0814 & 0.0086 &   0.0852 \\
      3 & 16.6118 & 16.6073 & 16.6077 &  0.1344 & 0.0046 &  -0.0275 \\
      3 & 16.6118 & 16.6097 & 16.6092 &  0.1131 & 0.0022 &  -0.0131 \\
      3 & 16.6118 & 16.6114 & 16.6086 &  0.1232 & 0.0005 &  -0.0029 \\
      3 & 16.6118 & 16.6123 & 16.6084 &  0.1144 & 0.0004 &   0.0026 \\
      3 & 16.6118 & 16.6266 & 16.6094 &  0.1135 & 0.0148 &   0.0891 \\
     4 & 19.7392 & 19.7249 & 19.7381 &  0.1586 & 0.0143 &  -0.0725 \\
     5 & 24.4161 & 24.4018 & 24.4164 &  0.2253 & 0.0142 &  -0.0583 \\
     5 & 24.4161 & 24.4205 & 24.4136 &  0.2217 & 0.0044 &   0.0180 \\
    \bottomrule
    \end{tabular}
  \end{table}

\subsection{Hydrogen like problem}

The calculation of hydrogen-like atomic potential lays a certain foundation for the calculation of atomic molecular structure. Under atomic units, the hydrogen atomic potential has the following form.

\begin{equation}\label
    & V(r)=-{1\over r}
    \end{equation}

where $r=\sqrt{x^2+y^2+z^2}$ is the radius. The boundary condition term attached to the network is $e^{-{r\over \nu}}$, where $\nu$ is the parameter to be trained.

The real wave functions and energy levels are:

$$\Psi_{nlm}^{R}(r)=N_{nl}e^{-{r\over n}}r^lF(-n+l+1,2l+2,{2r\over n})Y_{lm}(\theta,\phi),E_n=-{1\over {2n^2}}$$

where $F(-n+l+1,2l+2,{2r\over n})$ is confluent hypergeometric function, and $N_{nl}={2\over{a^{3/2}n^2(2l+1)!}}\sqrt{{(n+l)!}\over{n-l-1}!}$ is normalization coefficient.

The first 5 wave functions of the hydrogen atomic potentialshown in the Figure.\ref{fig.psi.H.3d}, the convergence curve is shown in the Figure.\ref{fig_3Hy:subfig:IE}, see Table.\ref{table.E.H.3d} (in $\hbar^2 \over {2\mu}$):

\begin{figure}[ht]
  \centering
  \subfigure[$E_1=-0.5000$]{
    \label{fig_3Hy:subfig:S00}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_0_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_2=-0.1241$]{
    \label{fig_3Hy:subfig:S01}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_1_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_2=-0.1244$]{
    \label{fig_3Hy:subfig:S02}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_2_A1.pdf}}
  \hspace{-2em}

  \subfigure[$E_2=-0.1242$]{
    \label{fig_3Hy:subfig:S10}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_3_A1.pdf}}
  \hspace{-2em}
  \subfigure[$E_2=-0.1234$]{
    \label{fig_3Hy:subfig:S11}
    \includegraphics[scale=0.3]{picture/H_3d/psi_3d_4_A1.pdf}}
  \hspace{-2em}

  \subfigure[Energy Eigenvalue]{
    \label{fig_3Hy:subfig:IE}
    \includegraphics[scale=0.6]{picture/H_3d/eigen.pdf}}
  \hspace{-2em}
  \subfigure[Total loss function]{
    \label{fig_3Hy:subfig:IL}
    \includegraphics[scale=0.6]{picture/H_3d/Loss.pdf}}
  \hspace{-2em}
  \caption{Hydrogen potential wave function}
  \label{fig.psi.H.3d}
\end{figure}

\begin{table}\centering
  \caption{Potential energy level and error of hydrogen atom}
  \label{table.E.H.3d}
  \begin{tabular}{rrrrrrr}
    \toprule

    \makecell[c]{Energy \\Level} & \makecell[c]{Theoretical \\Value} & \makecell[c]{Predicted \\Value} & \makecell[c]{Average \\Energy} & \makecell[c]{Energy \\Uncertainty} & \makecell[c]{Absolute \\Error} & \makecell[c]{Relative \\Error (\%)}\\
    \midrule
      1 & -0.500 & -0.5000 & -0.5000 &  0.0004 & 0.0000 &  -0.0012 \\
      2 & -0.125 & -0.1234 & -0.1241 &  0.0045 & 0.0016 &  -1.3180 \\
      2 & -0.125 & -0.1231 & -0.1244 &  0.0057 & 0.0019 &  -1.4842 \\
      2 & -0.125 & -0.1231 & -0.1242 &  0.0064 & 0.0019 &  -1.5135 \\
      2 & -0.125 & -0.1225 & -0.1234 &  0.0240 & 0.0025 &  -1.9714 \\
    \bottomrule
    \end{tabular}
  \end{table}
\section{Theory and Method}

\subsection{Base Theory}

This paper starts from the two methods of PINNs and VMC, and uses the frequency principle (that is, the neural network training process is convergent from low frequency to high frequency\cite{F_Principle, xuDeepFrequencyPrinciple2020, xuTrainingBehaviorDeep2019}), and introduces a wave function frequency hypothesis (the higher the solution level of the static Schr\"{o}dinger equation, The more high-frequency components of the wave function, the more nodes, knots, and interfaces of the wave function) to obtain a new method for solving the problem of the eigenvalue of the static Schr\"{o}dinger equation which is a special partial differential equation. This method quickly find solutions that meet each energy level by minimality of the error of equation residuals and orthogonal normalization conditions at the same time, and then improve the convergence accuracy by minimality of energy uncertainty. The algorithm framework is shown in figure \ref{fig.epinns}, which can obtain multiple eigenstates and eigenvalues in a single training, and also supports adding known states to orthogonal conditions to achieve multiple trainings to obtain more states. The reliability of the hypothesis is proved by numerical experiments.

Any given static Schr\"{o}dinger equation $\hat{H}\psi(\vec{R})=E\psi(\vec{R})$, where $\psi(\vec{R})$ is the required solution, $\hat{H}=\hat{T}+\hat{V}$ is the system Hamiltonian quantity, and $E$ is the energy eigenvalue. And there are $T=-{1\over2}\sum_i^D{\partial^2\over\partial{x_i^2}}$, $\hat{V}=V(\vec{R})$. In order to solve the first N eigenvalues and eigenfunctions of the equation, N functions with parameters and satisfying the boundary condition can be designed as the proposed solution $\psi_i^\theta(\vec{R})=Net_i(\vec{R})\cdot\mathcal{B}c_i(\vec{R}), {i \in [1,N]}$, where $Net_i(\vec{R})$ is the universal approximation, $\mathcal{B}c_i(\vec{R})$ is the term that satisfies the boundary conditions.

Thus the residuals of the equation can be defined $$\mathcal{F}=\sum_{i=1}^N\int_{x \in \Omega} \big| {\hat{H}\psi_i(\vec{R})-E_i\psi_i(\vec{R})}\big|^2d{\vec{R}}$$

where $\Omega$ represents the domain and the residuals of the equations represent the error between the proposed function and the real laws of physics;

Then based on the known data (including boundary conditions and other data) can be defined $$\mathcal{D}=\sum_{i=1}^N\int_{x \in \Omega_b}\big|\psi_i^{\theta}(x)-\psi_i(x)\big|^2dx$$

Where $\psi(x)[x \in \Omega_d]$ represents known data, the total $\mathcal{D}$ represents the error between the proposed function and the existing supervision data.

Knowed that the mutual orthogonal relationship should be satisfied between the eigenvest functions can be defined $$\mathcal{O}=\sum_{i=1}^N\sum_{j=i+1}^N\|\langle{\psi_i^{\theta}}|\psi_j^{\theta}\rangle \Vert $$

In order for the solution to be nonzero, each eigenwave function should satisfy the normalization condition, there is $$\mathcal{N}=\sum_{i=1}^N| \langle{\psi_i^{\theta}}|\psi_i^{\theta}\rangle-1 \Vert $$

To improve the eigenvalue convergence accuracy, and reduce the eigenvalue uncertainty, we can construct the eigenvalue uncertainty $$\Delta E = \sqrt{\langle{\psi_i^{\theta}}|\hat{H}^2|\psi_i^{\theta}\rangle-{\langle{\psi_i^{\theta}}|\hat{H}|\psi_i^{\theta}\rangle}^2}$$

In summary, we can construct the total loss function

\begin{equation}
  \label{Loss_total}
  \mathcal{L}=\alpha_1\mathcal{F}+\alpha_2\mathcal{D}+\alpha_3\mathcal{N}+\alpha_2\mathcal{O}+\alpha_2\Delta E
  \end{equation}

which can be proved if and only if $\psi_i^{\theta}(\vec{R})= \psi_i(\vec{R})$ and $E_i$ is the true value, $\mathcal{L}$ takes the minimum value of 0.

\begin{figure}[ht]
  \centering
  \includegraphics[scale=0.5]{picture/epinns.pdf}\\

  \caption{epinns principle, where $\Omega$ defines the field, $\mathcal{F}$ equation residuals, $\Omega_d$ is the known data coordinate domain of the equation solution, $\mathcal{D}$ is a data-driven term, generally containing boundary conditions, where $\psi_d(x)$ is the existing supervisory data, $\Delta E$ is the energy eigenvalue uncertainty, and $\mathcal{N}$ for the normalized conditional loss of the wave function, $\mathcal{O}$ is the orthogonal conditional loss. }
  \label{fig.epinns}
  \end{figure}

\subsection{Method}

The core of the numerical solution of the schr\"{o}dinger equation has several points: First is the construction of the proposed function; The second is the integral calculation which mainly involving sampling methods; The third is the algorithm of optimize. The entire algorithm of this paper consists of three parts: preprocessing including data sampling, training including the design of the proposed function and construction of loss function, and post-processing including error evaluation and related physical quantity calculation.

\subsubsection{Ansatz Function}

The construction of the ansatz solution function consists of two parts: one is the arbitrary approximation function, which mainly follows the idea of neural finite elements that is useing NN as a ansatz function which the input is the independent variable of the general differential equation and the output is the corresponding eigenstat of $Net^{\theta}_i (\vec{R})$. For boundary conditions, we can use the weighted margin method general operation that is making the term that satisfies the boundary condition $\mathcal{B}^{\theta}(\vec{R})$ embeds the solution function and trains its parameter variable.

Boundary condition items are divided into two categories: one is finite space boundary conditions, and the construction method of such boundary condition items is shown in Annex 1; The other type is infinity boundary conditions, which require a boundary asymptote term as a boundary term based on an equation.

\subsubsection{Calculation of Loss Function}

The integral of the ansatz function based on the neural network can only be used numerical integration, first of all, the loss function needs to be discretized, the sampling point is $X_s$, the corresponding weight of each point in the whole space is $\omega(X_s)$, and the corresponding weight of each point of the known data is $\omega_d (X_s)$, so there are discrete form of loss funtion.

Discrete form of residuals of the equation.
\begin{equation}
  \label{F_loss}
  \mathcal{F}={1\over S}\sum_{i=1}^N\sum_{s=1}^S\omega(X_s)\big| {\hat{H}\psi_i^{\theta}(X_s)-E_i\psi_i^{\theta}(X_s)}\big|^2
  \end{equation}

Discrete form of known data point error.
\begin{equation}
  \label{D_loss}
  \mathcal{D}={1\over S_d}\sum_{i=1}^N\sum_{d=1}^{S_d}\omega_d(X_d)\big|\psi_i^{\theta}(X_d)-\psi_i(X_d)\big|^2
  \end{equation}

Discrete form of orthogonal conditional.
\begin{equation}
  \label{O_loss}
  \mathcal{O}={1\over S}\sum_{i=1}^N\sum_{j=i+1}^N\sum_{s=1}^{S}\omega(X_s)\big|{\psi_i^{\theta}(X_s)}\psi_j^{\theta}(X_s) \big|^2
  \end{equation}

Discrete forms of normalized conditional.
\begin{equation}
  \label{N_loss}
  \mathcal{N}={1\over S}\sum_{i=1}^N\sum_{s=1}^{S}\omega(X_s)\bigg| |{\psi_i^{\theta}}(X_s)|^2-1 \bigg|^2
  \end{equation}

Discrete form of energy uncertainty.
\begin{equation}
  \label{Delta_E}
  \Delta E = \sum_{i=1}^N\sqrt{\langle\hat{H}^2\rangle-{\langle\hat{H}\rangle}^2}
  \end{equation}

where $\langle\hat{H}^2\rangle=||\hat{H}\rangle|^2={1\over S}\sum_{s=1}^{S}\omega(X_s)\bigg|H\psi_i^{\theta}(X_s)\bigg|^2$ and $\langle\hat{H}\rangle={1\over S}\sum_{s=1}^{S}\omega(X_s)\psi_i^{\theta}(X_s)H\psi_i^{\theta}(X_s)$

In order to solve multidimensional problems, this algorithm adopts a meshless method similar to DGM\cite{DGM} which is random sampling. There are two main sampling methods, one is gaussian sampling under infinite boundary conditions, and the other is uniform sampling under finite boundary conditions. Its purpose is to improve the efficiency of sampling. Because during the iterative solution of the model, each iteration needs to be resampled. Relying on the torch library can quickly generate the number of points needed directly on the memory of GPU, rather than first generating data on memory and then moving to video memory, which can reduce unnecessary overhead.

\subsubsection{Solving process}

The epinns solution process is similar to the conventional neural network training process, that is, it mainly includes space sampling, forward propagation, loss function calculation, key indicator storage, judgment convergence, backpropagation and other processes.The algorithm flow is shown in Algorithm (\ref{alg.train}).

\begin{algorithm}
  \caption{EPINNs Train process}
  \label{alg.train}
  \begin{algorithmic}[1]
  \REQUIRE Equations: potential functions, boundary condition terms. Network size. Training parameters, sampling form
  \ENSURE N energy level eigenvalues and wave functions to converge process data

  \STATE read excited wavefunction $\psi^{exit}(X)

  \WHILE{$epoch<=iterm$ and $\mathcal{L}<=\epsilon$}

  \STATE Sample in the defined domain$X_s$, sample the supervised data$X_d
  \STATE calculate $\psi^{\theta}(X_b)$, $\psi^{\theta}(X_s)$,$\psi^{exit}(X_s)$

  \FOR{$i=0$ to $10$}
  \STATE Using torch's automatic differentiation technique \cite{paszkePyTorchImperativeStyle}, the differentiation of $\psi^{\theta}(X)$ at $X_s$ is calculated
  \STATE Calculate $\mathcal{F}$ via Eq.(\ref{F_loss})
  \STATE Calculate the density matrix $P$ of each energy level according to equation Eq.(\ref{N_loss}) and Eq.(\ref{O_loss}) by $\psi^{\theta}(X_s)$ and $\psi^{exit}(X_s)$
  \STATE Calculate $\Delta E$ by $\psi^{\theta}(X_s)$ via Eq.(\ref{Delta_E})
  \STATE Calculate $\mathcal{N}$,$\mathcal{O}$ by $P$
  \STATE Calculate $\mathcal{D}$ by $\psi^{\theta}(X_b)$ via Eq.\ref{D_loss}
  \ENDFOR
  \STATE Calculate Total Lossfunction $\mathcal{L}$ via Eq.(\ref{Loss_total})
  \STATE Save model and datas
  \STATE Update various parameters with Ada
  \ENDWHILE
  \STATE Calculate other physical quantities
  \end{algorithmic}
  \end{algorithm}

The convergence criterion is that the eigenvalue is stationary and unchanged, and the two training wave functions are roughly unchanged. That is, the following metrics change less than the threshold.

 \begin{equation}\label
   & \epsilon = \sqrt{{1\over N}\sum_n^N|\lambda^{(t)}_n-\lambda^{(t-1)}_n|^2}+\sqrt{{1\over NS}\sum_n^N\sum_s^{S}\bigg|\psi_n^{(t)}(X_s)-\psi_n^{(t-1)}(X_s)\bigg|^2}
   \end{equation}

\subsection{Algorithmic practicability}

The practicability of algorithm includes several aspects, one hand is the calculation of physical quantities, the other hand is the mark of degenerate state, and the third is the error analysis of algorithm.

\subsubsection{Calculation of Physical Quantity}

The average value of physical quantities is calculated as follows

\begin{equation}\label
  & \overline{O} = {1\over S}\sum_s^{S}\omega(X_s){{\psi_n^*(X_s)\hat{O}\psi_n(X_s)}}
  \end{equation}

The average of the squares of physical quantities is calculated as follows:

\begin{equation}\label
  & \overline{O^2} = {1\over S}\sum_s^{S}{{|\hat{O}\psi_n(\vec{r}_s)|^2}\over P(\vec{r}_s)}
  \end{equation}

The uncertainty of physical quantities is there

\begin{equation}\label
  & \Delta O = \sqrt{\overline{O^2}-\overline{O}^2}
  \end{equation}

\subsubsection{State Labeling}

The state labeling algorithm is mainly aimed at the labeling of degenerate states. The general analysis indicates this kind of problem that requires finding a complete set of commuting observables(CSCO), and then looking for the eigenfunction of the CSCO. So that the eigen wave function is completely characterized. However, we only consider the energy eigenvalues when training, and the solved degenerate wave function is generally a linear superposition of the labeled degenerate energy levels. So in order to mark the degeneracy wave function, we can find the matrix representation of the CSCO in the mutually simplified wavefunction subspace, and then diagonize the matrix, and finally find the transformation matrix to achieve the mark of the wave function.

Taking the hydrogen atom as an example, it is known that the CSCO is ${\hat{H}, \hat{l}^2, \hat{l}_z}$, and the corresponding quantum numbers of the four degenerative energy levels of $n=2$ is ${l=0,m=0} and{l=1,m=0,1,-1}$. So you can first find $L$ which is representation of matrix $\hat{l}^2$ in the eigenfunction subspace of $n=2$. And then diagonally $L$ to get the eigenvalue and the eigenvector. For the state with $l=0$, there is obviously $l_z=0$, which gives the eigenform under that tag. For the other three states of $l=1$, $L_z$ of the matrix of $hat{l}_z$ is represented by using these three subspace of $L_z=0,1,-1$. Then diagonallyize the  $L_z$ to three states of $m= 0,1,-1$, i.e. fully marked these three states. In the end, the entire state of the hydrogen atom at the energy level of $n=2$ is completely labeled.

\subsubsection{Error Analysis}

Error analysis consists of three parts: First, the error comparison between the solved eigenfunction function and the real function; The second is the error analysis of the eigenvalue; The third is the error analysis of other physical quantities.

The precision analysis of the solution function is mainly to compare with the real solutions. But considering the degenerate state, the linear superposition of each true solution is also the true solution, so the greater probability solved wave function will be the linear superposition of the true solution. Therefore, the true solution corresponding to the neural network wave function can be set Eq.\ref{Real_Psi}.

 \begin{equation}
  \label{Real_Psi}
  \Psi_i^{real} = \sum_j^J C_ij\Psi_{ij},i\in[1,N]
  \end{equation}

Where $j$ is the degeneracy intrinsic function of the energy level $i$.

So just ask for the coefficient $C_ {ij}$ to compare it with the calculated neural network wave function, but the actual $\Psi_i^{real}$ is unknown, so you can use $\Psi_ {i}^{\theta}(X_s)$ to calculate $C_ {ij}$ according to the orthogonality theorem via Eq.\ref{C_ij}.

\begin{equation}
  \label{C_ij}
  C_{ij} = {1\over S}\sum_s^S w(X_s)\Psi_{i}^{\theta}(X_s)*\Psi_{ij}(X_s)
  \end{equation}

The final comparison of wave function errors will be achieved by comparing $\Psi_i^{real}$ and $\Psi_{i}^{theta}(X_s)$ and visualized in the main area.

The error analysis of the eigenvalue consists of $E_i$ and $\langle\psi_i^{\theta}|\hat{H}|\psi_i^{\theta}\rangle$.
$E_i$ is is the parameter given during the solution of the equation that represents the eigenvalue. The closer the two parts, the smaller the eigenvalue error Eq.\ref{delta_E}.

 \begin{equation}
   \label{delta_E}
   \delta E = |\langle\psi_i^{\theta}|\hat{H}|\psi_i^{\theta}\rangle-E_i|
   \end{equation}

Another term that indicates the average energy error is the energy uncertainty $\Delta E$ via Eq.\ref{Delta_E}, and the smaller the value, the smaller the error.

The error calculation of other physical quantities is mainly determined by the uncertainty of this physical quantity.

\section{Indroduction}

A key problem in quantum mechanics is to find the solution to the Schr\"{o}dinger equations. Its primary objective is to solve the eigenvalue problem of multi-dimensional static Schr\"{o}dinger equations. Furthermore, a high-precision and rapid solution of the multi-dimensional static Schr\"{o}dinger equation is of great significance to progress in quantum chemistry and computational materials science. The typical form of the multi-dimensional Schr\"{o}dinger equation is

\begin{equation}
    \hat{H}\Psi_n(\vec{R})=E_n\Psi_n(\vec{R}),
\end{equation}
where $\vec{R}$ is the $D$-dimensional coordinates with $\vec{R}={x_1, x_2, x_3,\ldots, x_D}$, $\Psi_n(\vec{R})$ is the eigenfunction of energy level $n$, $E_n$ is the eigenvalue corresponding to energy level $n$, $\hat{H}$ is the Hamiltonian amount of the system to be solved, and the general form is

\begin{equation}
    \hat{H}=-{1\over2}\sum^{D}_{i=1}{\partial^ 2 \over{\partial x_i^2}}+V(\vec{R}),
\end{equation}
where $V(\vec{R})$ is the potential function.

The static Schr\"{o}dinger equation cannot be solved accurately, except for a few models, such as infinite potential wells, harmonic oscillators, hydrogen atoms, Morse potentials, etc. The key process of approximate solutions consists of two parts, one is the representation method of the wavefunction, and the other is the model transformation and solving, i.e., it is generally necessary to convert the solution problem of the equation into an optimization problem, and then use the optimization algorithm to solve it. In general, the methods for solving the approximate solution to this problem include traditional algorithms and machine learning (ML)-based methods. The traditional algorithms include spectral methods, finite difference methods, finite element methods, etc. ML-based methods are generally a combination of neural networks (NNs) and traditional algorithms, i.e., they use a NN as a wavefunction ansatz to find the solution of the equation by minimizing an appropriate loss function.

The traditional Hartree-Fock (HF) method and density functional theory (DFT) \cite{martin2020electronic} are representations of the wavefunction using the basis function expansion method which is a spectral method and then use the variational principle to turn the equation solution problem into a constrained energy minimization problem, and then use the self-consistent field (SCF) iterative method to solve.\cite{kohn_sham_2021,martin2020electronic,lundqvist2013theory,Kohn_Sham_1965,Slater1972,Hohenberg_Inhomogeneous_Electron_Gas,Fermi_E,Thomas_atomic} The subsequent development of the quantum Monte Carlo (QMC) method solves the problem of energy integration in high-dimensional cases.\cite{excited_VMC, QMC-SJB} Traditional algorithms have a strict mathematical theoretical foundation, guaranteeing the solution's effectiveness. However, the system became unsolvable when solving the high-dimensional problem due to the curse of dimensionality even if the number of electrons increased slightly. Moreover, under the strongly correlated system, the above method introduces too many approximations and causes the solution to become unreliable. And the finite difference method\cite{}, finite element\cite{FEM_eigenvalues,HEUVELINE2003321} methods also suffer from the defects of being computationally intensive and difficult to deal with high-dimensional problems.

In recent years, with the development of ML, some numerical computation methods based on variational QMC and NN techniques have been gradually developed to calculate the multi-electron atomic and molecular static Schr\"{o}dinger equation, e.g., FermiNet\cite{FermiNet}, PauliNet\cite{PauliNet}, DeepWF\cite{DeepWF}, etc. These methods use a NN as the wavefunction ansatz, embed physical constraints (e.g., Pauli exclusion principle, boundary conditions, etc.) into the network by constructing a suitable form, then use the idea of variational QMC to construct the average energy as the loss function, and finally train the model to obtain an approximate wavefunction through a gradient descent algorithm.\cite{han_solving_eigenvalue_problems_2020} However, it still does not eliminate numerous limitations of traditional methods, i.e., the curse of dimensionality, low accuracy, and difficulties in calculating excited states.

Li et al.\cite{liNeuralnetworkbasedMultistateSolver2021} calculated multi-dimensional harmonic oscillators based on the variational Monte Carlo (VMC) method combined with NNs, which can obtain multiple degenerate energy levels simultaneously as network convergence. It has two improvements compared with the former approaches. One is to make the network output multiple functions as eigenfunctions of different energy levels, and another is to add orthogonal and normalized conditions in the loss function, which makes it possible to obtain multiple energy levels (including degeneracy)in a single training. However, the VMC-based method always has its significant disadvantages, i.e., the convergence accuracy is greatly affected by the integral accuracy. Further, adding the average energy to the loss function also contains two convergence characteristics. One is to make the solution satisfy the equation, and the other is to make the solution converge to a lower state, which seriously affects the convergence rate. The single training implementation by Li et al.\cite{liNeuralnetworkbasedMultistateSolver2021} gives multiple energy levels is also problematic. In particular, as the energy level of the single computation increases, the orthogonal conditions are increased by $N(N-1)\over2$, which reduces the convergence rate and causes some energy levels to interfere with each other.

Another class of algorithms is a series of methods developed by NNs combined with DFT\cite{moreno_deep_2020,DFT-ML}. Its main idea is to use molecular data to train NNs as exchange correlation energy, then calculate the electronic structure and other energies of molecules according to the traditional DFT algorithm. It has shown greater promise in molecular structure calculations.

At the same time, physics-informed neural networks (PINNs) developed in recent years based on finite element methods and NNs have shown great success in solving direct and inverse problems of partial differential equations. The relevant theories and applications in this field have gradually been enriched but mainly tend to solve definite problems in engineering. For example, good results have been obtained in Navier-Stokes equations, time-dependent Schr\"{o}dinger equations, etc.\cite{PINNs, DGM}. Its basic idea comes from the mean weighted residuals (MWR), i.e., using a fully connected NN as the solution ansatz, then finding the residuals of the equation on the space-time sampling point, and adding some known data (boundary / initial conditions) to the network, and finally the equation residuals and the weighted sum of the errors of the known point function as the final loss function, and trained by the stochastic gradient descent(SGD) algorithm to obtain the required solution by minimizing the loss function. In principle, such algorithms can also solve eigenvalue problems, but they cannot determine the energy level of the function they solve, nor can they reliably give other related energy levels.

In order to fast find wavefunctions of multiple eigenstates and energy levels of multi-dimensional static Schr\"{o}dinger equations with high precision, this paper gives a general algorithm for solving multi-dimensional static Schr\"{o}dinger equations based on the ideas of PINNs and VMC. It also gives a training method for solving multiple eigen wavefunctions and energy levels for multiple solving times. In addition, establish a specific evaluation method for solving such problems. The method only requires the equation form and boundary conditions to converge the eigenfunction and eigenvalue of the equation from low to high. Preliminary numerical experiments were completed in the infinite potential well, multi-dimensional harmonic oscillator, and hydrogen-like atom problems, which confirmed the general validity of the method.

The method opens up a new path for solving the multi-electron atomic and molecular static Schr\"{o}dinger equation and other high-dimensional eigenvalue problems. It also provides a new way of thinking for solving problems in the waveguide or resonator, which include various boundary conditions.

\section*{}