latex画一个三层bp神经网络的推导图

以下是一个简单的三层BP神经网络的推导图，其中包括输入层、隐藏层和输出层。

\begin{center}
\begin{tikzpicture}[x=1.5cm, y=1.5cm]

% 输入层节点
\node[circle, draw, thick] (x1) at (0,0) {$x_1$};
\node[circle, draw, thick] (x2) at (0,-1) {$x_2$};
\node[circle, draw, thick] (x3) at (0,-2) {$x_3$};

% 隐藏层节点
\node[circle, draw, thick] (h1) at (2,0) {$h_1$};
\node[circle, draw, thick] (h2) at (2,-1) {$h_2$};
\node[circle, draw, thick] (h3) at (2,-2) {$h_3$};

% 输出层节点
\node[circle, draw, thick] (y1) at (4,-0.5) {$y_1$};
\node[circle, draw, thick] (y2) at (4,-1.5) {$y_2$};

% 连接线
\draw[->, thick] (x1) -- (h1);
\draw[->, thick] (x1) -- (h2);
\draw[->, thick] (x1) -- (h3);
\draw[->, thick] (x2) -- (h1);
\draw[->, thick] (x2) -- (h2);
\draw[->, thick] (x2) -- (h3);
\draw[->, thick] (x3) -- (h1);
\draw[->, thick] (x3) -- (h2);
\draw[->, thick] (x3) -- (h3);
\draw[->, thick] (h1) -- (y1);
\draw[->, thick] (h1) -- (y2);
\draw[->, thick] (h2) -- (y1);
\draw[->, thick] (h2) -- (y2);
\draw[->, thick] (h3) -- (y1);
\draw[->, thick] (h3) -- (y2);

% 偏置节点
\node[circle, draw, thick, fill=gray!20] (b1) at (0.5,0) {$b_1$};
\node[circle, draw, thick, fill=gray!20] (b2) at (0.5,-1) {$b_2$};
\node[circle, draw, thick, fill=gray!20] (b3) at (0.5,-2) {$b_3$};
\node[circle, draw, thick, fill=gray!20] (b4) at (3,-0.5) {$b_4$};
\node[circle, draw, thick, fill=gray!20] (b5) at (3,-1.5) {$b_5$};

% 偏置连接线
\draw[->, thick] (b1) -- (h1);
\draw[->, thick] (b2) -- (h2);
\draw[->, thick] (b3) -- (h3);
\draw[->, thick] (b4) -- (y1);
\draw[->, thick] (b5) -- (y2);

\end{tikzpicture}
\end{center}

其中，$x_i$ 表示输入层的第 $i$ 个节点，$h_i$ 表示隐藏层的第 $i$ 个节点，$y_i$ 表示输出层的第 $i$ 个节点，$b_j$ 表示偏置节点，连接线上的数字表示权重。在训练过程中，权重和偏置会不断调整以最小化误差函数。