Hi Prof. B, around 9:20 you show a graphic on Eurika's work demonstrating how to directly solve for the eigenfunctions using sparse regression, where you want the number of xi coefficients to be as sparse as possible. Is there a way to know how many coefficients you need? Or is this another type singular value question where the number of coefficients you keep depends on how much information of your data is captured?
This is a great question! In general, we won't know how sparse the solution should be ahead of time. Typically we sweep through a parameter that controls sparsity and get a whole family of sparse models. Then we compare the performance of these on a data set that has been held out for testing. We are looking for the sparsest model that still has good predictive performance.
Hi Prof. B, at around 15:32, you show the point to represent the EDMD or DeepKoopman with cross-validation. I could understand that it will lead to small error but why less complexity compared to that without the validation?
I think it is because when you cross-validate, you are no more trying to fit all the measurement coordinates to the data and therefore, the complexity automatically decreases. But I have a question of my own, How do we cross-validate? let's say for eDMD model. One option is suggested in the video above which says we can divide the data into training and test data. But this will only work if the data is captured around a stable equilibrium. What if I want to obtain a linear model of a nonlinear dynamical system such as the cart-pend problem when the pendulum is in the inverted position and is being controlled. How do we cross-validate it then?
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I hadn't heard of the Koopman operator or Koopman spectral analysis before. Thank you for this fascinating introduction!
Hi Prof. B, around 9:20 you show a graphic on Eurika's work demonstrating how to directly solve for the eigenfunctions using sparse regression, where you want the number of xi coefficients to be as sparse as possible. Is there a way to know how many coefficients you need? Or is this another type singular value question where the number of coefficients you keep depends on how much information of your data is captured?
This is a great question! In general, we won't know how sparse the solution should be ahead of time. Typically we sweep through a parameter that controls sparsity and get a whole family of sparse models. Then we compare the performance of these on a data set that has been held out for testing. We are looking for the sparsest model that still has good predictive performance.
Hi Prof. B, at around 15:32, you show the point to represent the EDMD or DeepKoopman with cross-validation. I could understand that it will lead to small error but why less complexity compared to that without the validation?
I think it is because when you cross-validate, you are no more trying to fit all the measurement coordinates to the data and therefore, the complexity automatically decreases.
But I have a question of my own, How do we cross-validate? let's say for eDMD model. One option is suggested in the video above which says we can divide the data into training and test data. But this will only work if the data is captured around a stable equilibrium. What if I want to obtain a linear model of a nonlinear dynamical system such as the cart-pend problem when the pendulum is in the inverted position and is being controlled. How do we cross-validate it then?
That's a really good question, have you figured out?@@yashasvisai