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)?
The (perhaps naive) approach I've taken in the past is to search the space of ODE systems using genetic programming with a loss function obtained by integrating my candidate ODE system and measuring how closely the output matches the data I want to perform regression on.
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?
well it was my first thought, and I haven't had my coffee yet.
rsfmri data comprise multiple timeseries sampled from voxels throughtout the brain. The dynamics are complex, and there have been attempts to examine it through the lens of attractors etc. I'm not an expert in chaotic analysis, but will say that most of the advances in the field of neuroscience come from innovative analytic methods.
Like I said, it was my first thought, but there is a whole subfield examining brain dynamics through these lenses.
That kind of chaos is about simple systems that are unpredictable in the long term. That Lorenz system is just a few equations you could write on a napkin. Whereas your brain has billions and billions of neurons. (You might blame the complexity on having a lot of parts.)
That said, another aspect of dissipative chaos is that a highly complex system like turbulence (10^28 or so atoms bouncing off each other) often can be described by an attractor that is rather low dimensional and could be modeled with a few equations. So you might have some neural signal that tools like that could make an interesting model of.
as long as the dynamics are relatively low dimensional, it should be possible to estimate the differential equation, even if it is observed in a high dimensional space
