Since you want a mathematical equation of what the neural network has learned. There is a way to get it by representing each layer in equation form.
I am assuming x is input Data,σ is the activation function. Weights and bias matrix are denoted as W and b. So based on three hidden layer architecture below is the mathematical equation of a neural network :
This is the closed form of the function which neural network has learned during training.
If you want to derive the exact form, there are ways to achieve it. One is Sensitivity analysis and there is one more method that uses symbolic regression on a trained neural network to find the underlying equation. I am also attaching the research paper link for your reference which discuss about this in details:
Let me know if you have any further queries.
