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I am currently doing research as a PhD student in theoretical physics. Currently we are calculating physical quantities described by coupled non-linear algebraic systems of equations. These equations have to be solved for different sets of parameters. As far as I know there are several iterative methods, for example Newton's method, to solve these. Unfortunately, we do not always find the solution for a given parameter set with it. I have little idea about neural networks so far, but as far as I understand they can be used as estimators for problems.

My question now is, could neural networks be used to solve these equations, respectively to approximate the solution as good as possible? I tried to find something about this, but unfortunately my search was unsuccessful. Perhaps the whole question is moot, because the iterative methods are the best that exist.

Thanks in advance for possible answers or hints

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No, neural networks cannot magically solve non linear equations, at most they can approximate their behavior, but then you still have to do a grid-search to find the solutions, so this would not help you whatsoever...

That's the reason why you use gradient method, either first order methods like Gradient descent, or second order methods, like Newton method

In poor words, you can consider to use NN when you what to predict something, and you know the right solution to a bunch of such prediction (so, if you have a bunch of non linear set of equations and you have the optimal solution, you might try to use NN to learn a mapping from your equations to the solutions, if it exists)

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