Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth. \[ e^{x^{2}}=128 \]

Step 1 ：Take the natural logarithm on both sides of the equation \(e^{x^{2}}=128\) to bring down the exponent on the left side.

Step 2 ：Solve for x by taking the square root on both sides. This will give us two solutions, one positive and one negative.

Step 3 ：The solutions to the equation are \(x = \boxed{2.203}\) and \(x = \boxed{-2.203}\), rounded to the nearest thousandth.