Top wo Nerae! (1988)

In the future, Earth is under attack by gigantic aliens. Noriko Takaya, daughter of a famous captain in the battle against the aliens, enlists in the military to combat the aliens, but must struggle with her memories, the pressures of training, the ridicule of her classmates, and herself to make herself ready to be a mech pilot. She is eventually partnered up with her rival/idol, and given control of one of two Gunbuster mechs, Earth's most powerful weapon to combat the aliens. With the fate of the earth in her hands, Noriko flies off to meet her destiny...