I came across a post about forecasting volatility with a neural network; Honchar "Neural networks for algorithmic trading. Volatility forecasting and custom loss functions". However, it is not clear to me how the architecture is chosen.
- I don't know, and the author doesn't say.
- I don't know, and the author doesn't even say what $f$ is.
- I don't know, and given points 1 and 2 above, I don't see the point in digging through the paper to see whether the author ever says.
I doubt this paper will be very helpful. But it lists the author's email address, so you could contact him directly.
There's an entire "How to predict the stock market with neural networks" cottage industry now that seems to have little to do with sound financial principles or rigorous time series analysis and everything to do with the current Deep Learning/Tensorflow hype. The blog post you mention seems to fall squarely within that category.
That being said, some serious work on forecasting volatility with NNets has been done. See for example:
- Xu, Q., Liu, X., Jiang, C., & Yu, K. (2016). "Quantile autoregression neural network model with applications to evaluating value at risk" - Applied Soft Computing
- Borovykh, A., Bohte, S., & Oosterlee, C. W. (2017). "Conditional time series forecasting with convolutional neural networks" - arXiv preprint arXiv:1703.04691