Machine learning methods for panel (longitudinal) data I have a panel data set, for example:

obj  time        Y       x1      x2
1    1         0.1     1.28    0.02 
2    1        0.11     1.27    0.01   
1    2        -0.4     1.05    -0.06  
2    2        -0.3     1.11    -0.02   
1    3        -0.5     1.22    -0.06  
2    3        1.2      1.06    0.11    


I`m new at ML and until recently I did not know that this is a special (panel) data type. I predicted the value of a variable $Y(t+1)$ by values $x_1(t)$ and $x_2(t)$ (time lag) using linear regression model and MLP. But now I read some information about panel data analysis and realized that the methods I used were not suitable. At the moment I found that fixed/random effects models are suitable for panel data analysis. So, I have several questions:


What other methods are correctly used to analyze panel data (I`m interested in neural network models)? I read that these methods must take into account the dependency between particular object values and previous occurring values of this object (
what is in models with fixed and random effects).
I also tried to use MLP by feeding 2D data to it. I divided the panel data into $k=time$ $quants$ $count$  2D blocks and passed this data to the MLP input. For example above, $k=3$, ($input$ $layer$ $size= 4 = 
number$ $of$ $predictors* block$ $objects$ $count$). In this case $batch$ $size=1$. 
If I make the $batch$ $size=2$ and feed the neural network with 1D data ($input$ $layer$ $size=2$ for example above, will there be any difference? In both cases, the weights of the neural network will be rebuilt after the observations on all objects are transmitted in one quantum of time.
Panel data = multi-object time series.
in other words you have time series problem (time) for different objects (obj) that you are trying to predcit (Y).
If I were you I would just dissect this problem and start thinking in terms of time-series + another discriminative column called obj. What approaches do you know there? Here is really cool and MODERN time-series tutorial, check it out.
Regarding NN, why are you trying to squeeze it in so hard? Let the data tell you what algo can model it. Personally given these 3 features NN is just too much and you can achieve similiar results with less complexity using some simpler/less expensive approaches.