Dear Hisham,

Your question is an interesting one and you can find a good example about fitting an NP data representing reflectance value of wavelengths for coastal area. These data were fit to a polynomial equation using Back propagation ANN. 

Please check the following publication on my researchgate site.

Sea water chlorophyll-a estimation using hyperspectral images and supervised Artificial Neural Network.

Indeed: the rule is provided by the equation and the association is provided by the pairs (x,Q(x)). This association can be described by a two-dimensional space, if x and  Q(x) are scalars, or by a more complicated space, if they are vectors. So given n such pairs it's possible to construct networks that can represent such associations. E.g. for a linear association a perceptron suffices, for a non-linear association a multi-layer network is necessary. For an example, cf. https://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15381-s06/www/nn.pdf

You're looking in the wrong direction.  You want to find solutions of your quadratic equation, you need to do numerical optimisation to find solutions.  This has nothing to do with neural networks or even machine learning, but is "just math".

For example, the paper http://www.maths.manchester.ac.uk/~higham/narep/narep347.pdf discusses your problem in great length.

No-he's interested, precisely, in the ``inverse problem'': he *does*  know X (and Q(X))- and wants estimators for the matrices A,  B and C. So he wants to construct a network that, by  representing the association defined by the pairs (X, Q(X)), can provide estimators for the matrices A,  B and C. 

I agree.  But it is "just" a quadratic equation in $x$.   If $x$ were not a matrix it would be straightforward to solve---secondary school maths.  For matrices we need to solve an optimisation problem, and his question is to find $\mathrm{argmax}_{A,B,C}  \| Q(x) - x^T A x+B x+C \|$.  The paper I referred to above investigates that, and discusses how to numerically solve this.

So that approach would find you good values (optimal? I did not check if the resultant problem is a convex optimisation, but I can well imagine that it is) for A, B, C.

If that's not really what Hesham H. Amin wants to do, but instead wants to use an NN to represent the matrices A, B, and C, then, well, a normal NN architecture won't help him much.  After all, an NN computes s(Wn s(... s(W3 s(W2 s(W1 x))))).  That is powerful enough to represent an approximation of the original equation, but in itself will not yield the matrices A, B, C.

One could, of course, use an NN to represent A, and one for B, and C's almost for free, and be happy with it by putting them together in the original equation to amass to a computational phletora.  Why anyone would want to do so beats me.

No, the network doesn't, just, represent the matrices-it represents the pairing, defined by the equation. It's, already, known that scalar, linear regression equations, y = a*x+b, are considerably more involved to solve for a and b, given pairs (x,y), drawn from some distribution, than to solve, for fixed values of the parameters a and b. This is a well worked  subject and a lot is known about it, so it's useful to learn the background first. Linear regression with scalars demands exploring the landscape of globally convex functionals, matrix-valued relations, typically, involve exploring the (rugged) landscapes of functionals of multiple extrema. All this is, of course, more than twenty years old, but does require refinement, depending on the application.  

Neural networks solve optimization problems. It's useful to test a method on a ``simple'' problem. If this *is* such a simple problem, it would be useful that people that make this claim, also, provide some *relevant* references.  Indeed, for matrix-valued problems much depends on, certain, properties of the matrices. 

thanks all for your replies

I need to use neural network as shown NOT for function approximation. I need to use as a step before using global optimization to simplify the equation parameters. For example, the A's should be nxn, but I need to make it simple as a diagonal (vector) one to guess about the function distribution.