I have the following problem for a personal project of mine: I am solving a system of two differential equations that has 4 changing parameters. The output are two vectors of numbers. Let's say I am only interested in the final entry of the two soluti0ons, so essentially I have 4 inputs and 2 outputs. What I mean by this is that every time I change the 4 coefficients I get different values in the two final entries of the two vectors of solution x and y.
I want to use linear regression to learn this relationship, by this I mean that I do not want to solve everytime the differential equations but I indeed want to use the linear model that I find by all the data that I have.
How would I formalize this? I am having trouble because my output is a vector so that suggests a multivariate regression model but my input are 4 integers, so I am not too sure on how to proceed.