Can I use neural networks for a problem (in description)? I am modelling a process with 4 input parameters x1 x2 x3 x4. The output of the process is 2 variable y1 y2that varies with length and time. 
I also have data from experiments basically recording the trends in the two output variables as I vary my input variable. 
So far I have only seen neural network examples which will take input x1 x2 x3 x4 t 
(t  is time) and predict y1 y2 at said time t (no consideration of location). I would however like to also like to see variation with length as well at a given time t [y1a y1b... y1z; y2a y2b... y2z] where (a, b...z) are location points at an incremental distance dh from the start.
Any help is appreciated. TIA
To answer the titular question first: Yes, of course you can. Whether a neural network can give you better results than a simpler model, however, depends on:


How complex/non-linear the relationship between your input and output variables is;
Whether the neural network you have specified is able to learn this relationship efficiently;
How much training data you have.


With the information you have provided in the question, there's really no telling how it will perform against a simpler model, so I would advice trying both. 

As for how your output varies over time and location, just include both in your model. In case of a linear regression model, accounting for spatiotemporal autocorrelation could be done with a mixed model using an appropriate covariance structure. The challenge for a neural network would be how to specify one that can learn this type of relationship (for starters, read up on RNNs for longitudinal data).