# How to deal with "real" data and closed formed equation in machine learning?

My goal is to perform regression over a set of data coming from the "real world" (sensors).

The data is in tabular format. There are 6 independent features with very different values (scaling is necessary). Additionally, the dependent variable has a large variability (potential scaling needed?).

An initial classical training shows weaknesses in the prediction when one of the variables (Let's say X1) is low in absolute value.

Experts tell me that the value to be predicted (let's call it Y) in this specific region (where X1 is low) can be approximated by a linear regression. So if all other features are constant, X1 and Y have a linear dependency of the type Y= a * X1 + b.

The problem is that coefficients "a" and "b" depends on the other features a = f(X2,X3,X4,X5), b=g(X2,X3,X4,X5)...

Note that I have a table of the coefficient "a" and "b" for several combinations of the 5 other features.

I would like to integrate the linearization that is "physics informed" into the training process. But how can I do that? I had a look on Physics Informed Neural Network, but they are oriented for PDE only and not closed form equation like the one I have.

For me, a natural thing to do would be to generate fake data in this area through the equation. Would it be considered as Physics Informed Machine Learning? I won't see the difference between adding "physics informed" data, and adding a loss which would try to fulfill the equation.

Many thanks,

• You can use mathjax to format your formulas.
– nbro
Commented Jun 27 at 11:33
• Thanks, for the tip ! Any other tip for my problem though ? Commented Jun 28 at 14:53
• Sorry I am not familiar with physics-informed ML and right now I don't have time to get informed about it. Maybe you can clarify what is meant by "physics-informed" in general and in your context, though, but I guess you also don't know exactly what that is because you're asking that question
– nbro
Commented Jun 28 at 23:24