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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.

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