$\because$一条抛物线的形状与抛物线$y=2x^{2}+3$形状相同，
$\therefore a=\pm 2$，
设抛物线的顶点式为$y=\pm 2\left(x-h\right)^{2}+k$，
由$y=-\frac{1}{2}(x+1)^{2}-2$可知顶点$\left(-1,-2\right)$
$\therefore$此抛物线顶点坐标是$\left(-1,-2\right)$，
$\therefore$抛物线的顶点式为$y=\pm 2\left(x+1\right)^{2}-2$.
故答案为：$y=2\left(x+1\right)^{2}-2$，$y=-2\left(x+1\right)^{2}-2$.

$\because$一条抛物线的形状与抛物线$y=2x^{2}+3$形状相同，
$\therefore a=\pm 2$，
设抛物线的顶点式为$y=\pm 2\left(x-h\right)^{2}+k$，
由$y=-\frac{1}{2}(x+1)^{2}-2$可知顶点$\left(-1,-2\right)$
$\therefore$此抛物线顶点坐标是$\left(-1,-2\right)$，
$\therefore$抛物线的顶点式为$y=\pm 2\left(x+1\right)^{2}-2$.
故答案为：$y=2\left(x+1\right)^{2}-2$，$y=-2\left(x+1\right)^{2}-2$.