There are a lot of resources for time series forecasting using neural networks. Because neural networks are so flexible, you can have it produce multiple outputs [x1, …, xt] or forecasts in the future. It requires a custom cost function that aggregates the loss for each prediction.
LSTM’s are popular for forecasting time series — this machine learning mastery tutorial is an example:
How to Develop LSTM Models for Time Series Forecasting - Machine Learning Mastery
Long Short-Term Memory networks, or LSTMs for short, can be applied to time series forecasting. There are many types of…
Current cutting edge (to my knowledge) is to use dilated CNN’s for forecasting. These are much faster to train than LSTM’s.
Time Series Forecasting with Convolutional Neural Networks - a Look at WaveNet
Note: if you're interested in learning more and building a simple WaveNet-style CNN time series model yourself using…