# How to find the relationship between one dataset and two others?

From a STEM problem, I vary a variable x within a range and calculate quantities $$U(x)$$, $$V(x)$$ and $$W(x)$$. I want to figure out an analytical relationship between unknown $$U(x)$$ and the other two quantities.

However, in my setup, I have 3 vectors: $$A(x), B(x)$$ and $$C(x)$$. Each vector results in a value of $$V(x)$$ and $$W(x)$$. This means that my $$V(x)$$ and $$W(x)$$ are in fact: $$V_A(x),V_B(x),V_C(x),W_A(x),W_B(x)$$ and $$W_C(x)$$ (where the subscripts indicate the vector used to calculate the quantity). However, to calculate $$U(x)$$, I must choose a pair of distinct vectors. So, my $$U(x)$$ are in fact $$U_{AB}(x), U_{AB}(x)$$ and $$U_{BC}(x)$$ (where the subscripts indicate the chosen pair of vectors).

I have reason to believe that each of my 3 $$U(x)$$ can be expressed entirely in terms of the 6 $$V(x)$$ and $$W(x)$$ terms. However, I am having trouble finding this relationship.

On top of that, $$U(x)$$ takes on integer values while $$V(x)$$ and $$W(x)$$ are smooth functions with only a few discontinuities. So, I tried various combinations of derivatives, sigmum functions, etc to try and recover $$U(x)$$ without explicitly calculating it. However, I am now out of ideas.

I am not too familiar with data science methods, and so wanted advice on how I could go about finding an analytic relationship between my quantities. I figured I should try to use machine learning (where the software could try various relationships until a match is found), but I haven't done that before. So, I was hoping for advice on a path to take to try and figure my problem out. I have access to supercomputing nodes, if it's anything.