解析：
由图可知,
$$
\begin{aligned}
& \mathrm{S}_{\text {阴影 }}=S_{A B C D}-2 S_{\text {扇形 }}, \\
& S_{A B C D}=2 \times 2=4, \\
&
\end{aligned}
$$
$\because$ 四边形 $\mathrm{ABCD}$ 是正方形, 边长为 2 ,
$$
\therefore A C=2 \sqrt{2} \text {, }
$$
$\because$ 点 $\mathrm{O}$ 是 $\mathrm{AC}$ 的中点,
$$
\therefore \mathrm{OA}=\sqrt{2} \text {, }
$$
$$
\begin{aligned}
& \therefore S_{\text {䂝忹 }}=\frac{90^{\circ} \pi(\sqrt{2})^2}{360^{\circ}}=\frac{\pi}{2}, \\
& \therefore S_{\text {阴影 }}=S_{A B C D}-2 S_{\text {扇形 }}=4-\pi,
\end{aligned}
$$
故答案为: $4-\pi$.