There are methods to account for uncertainty in the training data. If you can quantify it or estimate, then you can modelize the error but this has more to do with statistics  and with statistical analysis in general than with neural networks (I mean, the answer with DNNs will not be different than for any predictive model).
Also, you should know whether it’s your inputs, your outputs or both that are biased. Unfortunately, if your outputs are biased, then your model will be biased. If your inputs are biased, then your model might learn to cope with that (to some extend of course: the expression “garbage in, garbage out” still holds)