Back when I was at MIT and starting to think about the predictive capabilities of reservoirs, I wanted to pull out the dynamical systems textbook and answer all my questions about prediction.
The first dynamical systems textbook I pulled out was Strogatz. I realized that I could hit the problem of prediction with that textbook in the limit of weak input, when the basin-attractor portrait failed to change, and that resulted in this paper. The second book I looked up was one on "Random Dynamical Systems". I got one chapter in when I realized something-- this book wasn't answering any of my questions on prediction. I realized that a conceptual shift was needed. These dynamical systems were not "random". They were filters of input, and the input was the signal, and it was incorrect for all the problems I was working on to treat the input as noise. I barely care about what the state of the system is; I only care about how the state of the system relates to the past of the input, something that may be harder to keep track of. The fix to this, I think, is to look at the joint state space of predictive features of the input and the state of the system and find dynamics on that joint state space. I did this in this paper. You have to know something about the input. The math gets a bit complicated, but I'm hoping that slight fixes to the Strogatz textbook can be imported in the more general case! Altogether, I think a new textbook on dynamical systems with input is in order, one that includes more recent work on reservoir computing. These input-dependent dynamical systems actually do a computation, and so many fields-- from biophysics to theoretical neuroscience-- care about quantifying exactly how well that computation is done. Considering the input as noise is the opposite of solving the problems in these fields. I think this is a classic example of how bio-inspired math could spark an entirely new textbook.
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November 2022
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