Map vector to vector with neural network or other black box model

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I want to find a mapping function which can map two vectors:
The size of x is 3 and y is 4.
There're nonlinear realationships between these 2 vectors, which is
It's that clear it's impossible to get y analytically.
Since I'm new to machine learning, I wonder if it is possible to use neural networks or other black box models to do this mapping, and what kind of models would you recommend more?
(For those of you interested in specific scenario: This is the conversion of parameters between two induction motor steady state model, inverse Γ model and T model. There are redundancies in T model, and inverse model is the simplified model without redundancies.)

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