Example 12 Given $x, y, z \in(-1,1)$, and $x y z=\frac{1}{36}$, try to find the minimum value of the function $u=\frac{1}{1-x^{2}}+\frac{4}{4-y^{2}}+\frac{9}{9-z^{2}}$.
Given that $x^{2}, \frac{y^{2}}{4}, \frac{z^{2}}{9} \in(-1,1)$.
$$
\begin{array}{l}
=3\left(1+\frac{1}{36}+\frac{1}{36^{2}}+\cdots\right)=3 \cdot \frac{1}{1-\frac{1}{36}}=\frac{108}{35} \text {. } \\
\end{array}
$$

When and only when $x^{2}=\frac{y^{2}}{4}=\frac{z^{2}}{9}=\frac{1}{36}$, $u$ takes the minimum value $\frac{108}{35}$.