Elbow grease and a scotch brite in the low abrasive. Red, I think.

low abrasive such as barkeeper's friend ?

I would reach out to Samsung first, to avoid doing further damage.

Are there any directions in the literature that came with the cooktop, which suggests what to use for cleaning?

Just as Joseph just stated. You have ruined the top's finish. Now you need to reapply the brushed stainless finish. Which is the beauty of stainless as this can be repolished.

I assume that would also remove the markings/labels? You can order a new "chassis", I think? Check PartsWarehouse or similar....

I would try dilute phosphoric acid (with all safety precautions observed) between first trying Barkeeper's Friend and third trying heavier grit sanding. Depending on cost and the value of one's time, replacement also seems to be a valid approach, and would allow experimenting with the old surface without tying up the cooktop. However, replacement may take a lot of time itself, possibly requiring disconnection of the gas line with attendant risks.

Sorry for your problems. I hate my ceramic/glass cooktop and was longing for a metal/enamel one. I guess all types have their issues.

@Louise Smith...I have an enameled Thermador gas rangetop, now almost 5 years old. It is easy to clean, especially if you don’t let it go for weeks before getting to the mess. Usually, just a Clorox wipe or two, and a microfiber cloth, does the trick. At times, a drop of Softscrub with a damp cloth is required.

My DH, who has been in charge of the kitchen since my lung transplant, rarely gets to the rangetop timely. Two weeks can go by (gross, I know). For that kind of buildup, he uses Weiman Stainless Steel Cleaner and Polish, and all is brought back to normal..bright, and squeaky clean, with no residue left behind.