Shortest path using Artificial Neural Networks

I am working on my Master Thesis

My equations are of form Ax=b

Inputs: For every A (mxn matrix) there are different b vectors

Outputs: For every b we have x vector and a scalar

Example

Input....................: Outputs

A(mxn); b1 (mx1): x1[vector], +a1

A(mxn); b2 (mx1): x2[vector], +a2

A(mxn); b3 (mx1): x3[vector], +a3

.................................

.................................

.................................

B(lxk); c1 (lx1): y1[vector], +d1

B(lxk); c2 (lx1): y2[vector], +d2

B(lxk); c3 (lx1): y3[vector], +d3

.................................

.................................

I want to train these set with neural networks and see if i am getting valid output vector and scalar for every other given Matrix H (*x*) and vector e(*x1) belongs to the same kind of problem.

Any Help would be appreciated. Thank you!

Best Regards, Veeranki