How to predict tf-Idf vector with Neural Network?
I hope you do understand, that trying to predict some abstract TF-IDF vector based on read article you are trying to teach a neural network to produce a new article for a given user. It would be a major step in artificial intelligence if achieved, so keep us posted on your progress.

However, if you would think of alternative approaches, I would suggest following steps to make your neural network to work:


You need a classification/clustering model for your articles (which can be based on your TF*IDF, or on some tags assigned to articles). The more classes you could squeeze from your model, the better would your resulting recommender be. On that step, you should be at least able to provide user with articles similar by topic.
You will need at least two-hop data set of user reading current article and hopping to the next one which would be used as a training set for actual predictor. Here, you will train a neural network (or any other model) which will try to predict a next-article class (from step one) based on current article - that would be a recommender system.


Of course, you may, given you have data from (2), train a neural network that would answer the question of "If user reads article A, will the next article contain word B", but that would require you to have the size of output variables of your TF*IDF matrix (for any reasonable dataset that would be at least 20-30k words unless you work with books for toddlers) which makes your model overcomplicated (that would be, actually, a TF*IDF matrix size number of models for every word).