Thanks so much, this makes a lot of sense and this is precisely what I want:

adding these class weights to your loss function will train
your network to more often correctly label class-0 samples as
class-0 (fewer false negatives)

Out of curiosity, do you mind elaborating on this problem and what one might do?

Just to emphasize what I alluded to above, this depends on what
you mean by “getting the 0s right” and “punish mistakes for the 0s”.
If you consider mislabelling a class-1 as a class-0 (a false positive)
as not “getting the 0s right” and a “mistake for the 0s,” then this
likely won’t accomplish your goal.