Admittedly I've only had time to skim the paper once, but I'm having some trouble understanding what the neural network is doing here--is it determining the order of the system (the number of first-order ODEs we need to represent it)?
What they're doing here is not that. I think if I understand it correctly they're using the neural network to generate data which when symbolically regressed with PySR yield the RHS of each ODE in the system.
What's not immediately obvious is what the benefit of introducing the neural network is--does it make it faster than the "direct" naive approach?