P.A.X. is the first boss in the game, available at the end of Zone 1 after restoring power to the area and returning to the train station. Once you reach the area, you will be locked in.
The mechanics are pretty easy here. There are two different bars. He has a health bar (red) and a "shield" type bar (yellow). Always stay close to him, and under his feet, attacking when you can. Watch out for the stomps, and continue hitting him until you fill the yellow bar. Once the yellow bar fills, you will launch a rocket barrage. During the barrage, you must stay under him in order for him to hit himself, damaging himself enough to become vulnerable to attacks. Attack him during his "kneel" phase for 1/3 damage to his health (red) bar. He will pull out a sword during the 2nd and 3rd phase, continue staying under his feet, repeating the same method to take him down.
Make sure you maximizing the amount of Vital Injections + Vital Boosts implants in your suit!