How to train this neural network? I seek the neural network (NN) which satisfies the 100 equations (i=1,2...100)   

$\sum_{j=1}^{2000} NN(A_{ij},B_{ij},C_{ij})=Q_i$.

Where A,B,C are 100x2000 matrices

So I know Q, A, B and C
How can one find the NN? (ie train it using this data)
I have a second datset to test it on.
I also have MATLAB neural network toolbox, but that is not a necessity for me to use that.
Let $w$ denote the weights of the neural network.  Define the objective function $\Psi$ by

$$\Psi(w) = \sum_i \left(\left(\sum_j NN_w(A_{ij},B_{ij},C_{ij})\right) - Q_i\right)^2.$$

Then, find $w$ that minimizes $\Psi(w)$ using gradient descent.  You can find the gradient of $\Psi(w)$ using backpropagation through the neural network.  You can use all the standard methods for speeding up training of neural networks: stochastic gradient descent, Adam or momentum or Adagrad, etc.

This is very similar to training neural networks for regression problems.  The only difference here is that you are summing the output of the neural network on multiple inputs.  It is straightforward to adjust the objective function to take that into account, as shown above.