Example 4 Find all integer pairs $(x, y)$ that satisfy the equation $y^{4}+2 x^{4}+1=4 x^{2} y$.
Solution: Transform the original equation into
$$
\begin{array}{l}
2 x^{4}-4 y x^{2}+\left(y^{4}+1\right)=0 . \\
\text { Then } \Delta=(-4 y)^{2}-8\left(y^{4}+1\right) \\
=-8\left(y^{2}-1\right)^{2} \geqslant 0, \\
\left(y^{2}-1\right)^{2} \leqslant 0 .
\end{array}
$$

Thus, $y^{2}-1=0$, i.e., $y=1, \cdots 1$.
When $y=-1$, the equation has no solution;
When $y=1$, $\left(x^{2}-1\right)^{2}=0, x=1$ or -1.
Therefore, all integer pairs satisfying the original equation are (1,
$$
\text { 1), }(-1,1) \text {. }
$$