I am working with a reduced-order mathematical model for the motion of a offshore platform.

The deduction of equations of motion requires a lot of symbolic computation as differentiation of time-dependent variables and inversion of symbolic matrices. I was using Wolfram Mathematica with this purpouse. What packages can be used for this type of algebraic manipulation in Julia?

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You might want to check out this workshop which does this in full using the universal differential equation methodology:

DataDrivenDiffEq.jl is the library to look at for doing this:

https://datadriven.sciml.ai/dev/sparse_identification/sindy/

and the example mixes neural networks into the training methodology can be found here: