Sorry to say, but there is no way to remove scratches, especially deep ones, from stainless steel. Any attempt to remove them by sanding or using an abrasive such as steelwood (Brillo or SOS pads) will produce a larger area of fine scratches that will be even more noticeable.
The best approach is to use a stainless steel cleaner on a regular basis, which will improve the overall lustre of the sink, taking attention away from the individual scratches. My favorites are Barkeeper's Friend, ZUD and Bon Ami.
Take care
