The SINDy Algorithm

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We cover the SINDy algorithm of Nathan Kutz and Steve Brunton, and we give our own take on the algorithm. Our approach retains the advantages of the original algorithm while also being more robust to noise. Specifically, we perform a weak version of the algorithm (Published in CDC2019), and this approach simultaneously computes the projection of a dynamical system onto a span of basis functions via the inner product given in the previous lecture.

This approach simultaneously computes the projection on Koopman generators through the dynamical system inner products developed in the previous video.

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Very kind of you to share the MATLAB code. It really helps to understand the idea.

何何-vc

Welcome back! Glad to hear you're not dying!

bengski

Thanks for sharing this. I had seen the SINDy algorithm before, but this is the first time that it clicked that the algorithm is really all about the Liouville operator! With the general prominence of weak-formulation based methods and the popularity of the SINDy algorithm, I'm surprised that this is the first that I've encountered a weak-formulation based approach to the SINDy algorithm.

Two questions. First, regarding the relationship between kernels and observables: the takeaway as I understand it is that if the g_i are such that K(x, y) = \sum_{i} g_i(x)g_i(y) defines a "good" kernel, then these are probably good choice of basis for your space of observables. If we start with a kernel that we like, how would we get these observables? Could you get the from some kind of spectral decomposition of the kernel function?

Second, you say in your video on sampling theory that L^p spaces don't mesh well with modeling based on sampled trajectories because functions are, in principle, allowed to vary on any set of measure zero. On the other hand, you present a weak formulation of the SINDy algortihm (which works off of sampled trajectories), and weak formulations of ODEs/PDEs are typically framed in terms of Sobolev spaces. So, what is the underlying function space for the weak SINDy method? Do you break the usual trend by posing a weak formulation of a problem over a space of continuous functions, or are you somehow working with L^p spaces in the background?

By the way: you seem to have one of the video chapters mislabeled - the Matlab example is at 17:58 rather than 7:58.

bengski

Suppose we obtain the dynamics of a system from time series data with amazing SINDY. Now we want to cross validation for dynamical system. How are the initial values of the states selected? make a video about cross validation please! tnx

miladyazdanpanah

It would be very cool to have this incredible work in pySINDy as well :)

JoaoAntonioCardoso

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