Example 3 Find all integer pairs $(x, y)$ that satisfy the equation $y^{4}+2 x^{4}+1=4 x^{2} y$.
(1995, Jiangsu Province Junior High School Mathematics Competition)
Analysis: The original equation can be transformed into $2 x^{4}-4 y x^{2}+\left(y^{4}+1\right)$ $=0, \Delta=16 y^{2}-8\left(y^{4}+1\right) \geqslant 0$, then $\left(y^{2}-1\right)^{2} \leqslant$
0. Therefore, $y^{2}-1=0, y= \pm 1$.
When $y=1$, $x= \pm 1$;
When $y=-1$, $x$ has no real solutions.
Thus, $(x, y)=( \pm 1,1)$.