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<|diff_marker|> ADD A1000 user

I have been given some data points (x,y) which are generated using an unknown function f(x). I am required to fit this function f(x) using least squares method.
I have used least squares method for linear regression but now I want to try other methods like polynomial regression etc.
My question is how can I choose which type (linear/polynomial/logarithmic/exponential etc.) regression should be used?
Is there any algorithm/technique which automatically chooses best fitting function?
I tried looking into machine learning libraries like scikit-learn but didn't find anything useful.
Can someone please help me out?

I am trying it using python language but if anyone has any idea regarding matlab then please share that too!

Edit: I found out about scipy.optimize.curve_fit() which takes arbitrary function f(x) so we can define our own function too! But my question still remains same i.e how can we choose best fitting function? Is there any algorithm/technique which automatically chooses best fitting function?

Edit: It seems like people are suggesting "try all possible functions" approach but it doesn't seem feasible because there are infinite number of possible functions! So what should be done in such cases? Can anyone suggest some good references where such techniques are discussed?

Edit: To give more context about my problem - I am trying this on data generated by an unknown simulation code (which has many parameters). So basically I want some sort of inverse modeling technique where given data points (x,y), we find out model parameters that generate similar data points. In short I want an automated way/modeling technique which tries different models/functions until it finds one which fits given data well!

Edit: The data that I am working on is generated by simulating physical phenomena (so its non-linear). Also note that x values are discrete here!

Edit: The main reason why I am asking this question is because - when I tried polynomial regression using degree=2 , it didn't fit very well whereas when I tried degree=10 , it fit almost perfectly! But when i tried higher degrees (>10) then again error started increasing! So clearly degree=10 was best choice here but how do we know beforehand without trying all possibilities? And also what if instead of polynomial regression we had tried exponential or logarithmic etc.? How do we know beforehand without trying all possibilities?

Edit: This might sound silly but if there exists a mathematical solution then why do people even talk about machine learning techniques at all? Isn't ML just another black box technique where input goes in one end & output comes out from other end without us knowing much about whats happening inside? If there exists some mathematical solution then why don't people use that instead ML?

Edit: Thanks everyone for helpful answers/comments!
I will try Bayesian Optimization approach suggested by @YvesDaoust & @Shrikant Shende!
Also thanks @Gertjan for providing references!