How to approach mapping families of vectors on a lattice and forecast resulting value
It really seems like a problem of formulating your probkem. On your 2D exemple you can split your $X_i$ along your axes and juxtapose them to build a data set. Each $X_i$ has two components : along axis 1 ($X_{i1}$) and 2 ($X_{i2}$).

Basically you would just build your data set by juxtaposing the 18 columns :

$X_{11}$ , $X_{12}$ , ..., $X_{i1}$ , $X_{i2}$ ,..., $X_{91}$ ,$X_{92}$ 

To try to predict your y. 

You can augment the dimension of you problem by similar juxstaposition of your features. 
It would get a bit more complex if you augment the dimension of y, as you would need multiple models. 

Trying to predict $y_{t+1}$ would lead to other considerations, like considering $y_t$ as a feature of your model.