Time series forecasting: from ARIMA to LSTM

The combination of differential equations (e.g. ODE of SIR models) and HMM are often used in epidemiology. The hidden states are models as ODEs and the observation process are modeled as HMM. One example is pomp. The model is trained on existing data and produces forecasting on the future. Another goal of this kind of model is to understand epidemiology related parameters. More examples can be found in here and this book

There are a couple of good review papers on the topic of deep learning for forecasting:

• Neural forecasting: Introduction and literature overview
• Recurrent Neural Networks for Time Series Forecasting: Current Status and Future Directions
• And a very good presentation by the amazon team

A word of warning though: I am a very big fan of LSTM based forecasting and I advocate for it alot in my various roles. But I would be the first to tell you to tread very, very carefully: The number of use cases where LSTM provide an advantage over traditional statistical models is very limited, and Deep Learning is very far from being an established theoretical topic, the way ARIMA or State Space Models are.

The "classical" methods comprise much more than ARIMA and GARCH (which address different questions, and at least ARIMA is not very useful for forecasting), e.g., decomposition, Exponential Smoothing etc. I recommend this very good free online textbook by Athanasopoulos & Hyndman.

I agree that there is very little in terms of textbooks on HMMs or NNs as used for forecasting, and I would be interested in any pointers.

Looking at book reviews in the International Journal of Forecasting may be helpful (even though the list of search results is admittedly not).