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

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

Dilated neural networks for time series forecasting - Strata Data Conference in San Francisco 2019