2. Let $P$ be any point inside a regular tetrahedron $ABCD$ with edge length $\sqrt{2}$, and let the distances from point $P$ to the four faces be $d_{1}, d_{2}, d_{3}, d_{4}$. Then the minimum value of $d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ is $\qquad$
2. $\frac{1}{3}$.

From $V_{\text {tetrahedron } A B C D}$
$$
\begin{array}{l}
=V_{\text {trilateral pyramid } P-A B C}+V_{\text {trilateral pyramid } P-D B C}+ \\
V_{\text {trilateral pyramid } P-C D A}+V_{\text {trilateral pyramid } P-D A B} \\
\Rightarrow \frac{\sqrt{2}}{12}(\sqrt{2})^{3}=\frac{1}{3} \times \frac{\sqrt{3}}{4}(\sqrt{2})^{2}\left(d_{1}+d_{2}+d_{3}+d_{4}\right) \\
\Rightarrow d_{1}+d_{2}+d_{3}+d_{4}=\frac{2 \sqrt{3}}{3} \text {. } \\
\end{array}
$$

By the Cauchy-Schwarz inequality, we have
$$
\begin{array}{l}
d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2} \\
\geqslant \frac{\left(d_{1}+d_{2}+d_{3}+d_{4}\right)^{2}}{4}=\frac{1}{3},
\end{array}
$$

with equality if and only if $d_{1}=d_{2}=d_{3}=d_{4}=\frac{\sqrt{3}}{6}$.