training neural net with multiple sets of time-series data
Yes, this is a straightforward application for neural networks.  In this case yk are the outputs of the last layer ("classifier"); xk is a feature vector and yk is what it gets classified into.  For simplicity prepare your data so that N is the same for all.  The problem you have is perhaps that in the case of time series you won't have enough data: you need (ideally) many 1000's of examples to train a network, which in this case means time series, not points.  Look at the specialized literature on neural networks for time series prediction for ideas on network architecture.

Library: try Pylearn2 at http://deeplearning.net/software/pylearn2/  It's not the only good option but it should serve you well.