I have several published papers on the so-called Koopman-von Neumann Hilbert space representation of Hamiltonian mechanics, yet I know very little about the Koopman operator theory. I feel like there is a language barrier that makes it hard for me to study this topic, so I'm glad that these videos exist.
One relatively minor point from the control theory point of view I’d say regardless of nonlinearity if the control scheme is convex, then we will be able to implement a nice controller for the most part. The big challenge is if the model predictive control scheme has to solve a non-convex optimization problem to find even a good control input. If we are in the non-convex regime we don’t have good algorithms that can operate without babysitting, so it becomes untenable for many control applications. Therefore, non-convexity is the challenge rather than non-linearity.
Thank you for the great talk! Minor issue: I believe there's a typo at @15:47 : phi_lambda should be x_2, and phi_2mu should be x_2 + (lambda - 2mu)/lambda * x_1^2
At about 23:11, you mention a pseudoinverse that you get by regression. Is an inverse covariance matrix used, and if so, does it follow from the data? Excellent intro and I look forward to the other lectures.
At about 20:52 you derive \phi(x)=e^{-1/x}. You can just as easily show that e^{-\lambda / x} has eigenvalue \lambda. So there's a continuous spectrum, and 0
One more question. The Koopman operator is the adjoint of the Frobenius Perron operator, the latter advancing a density and the former advancing a scalar (an observable.) Is it possible to repeat all the Koopmanism stiuff with the F-P operator? Also, in the conventional use of adjoint methods, you can either advance the F_P forward in time or the K backward, taking the inner product of the scalar and the density at the initial or the final time. Does any of this appear in Koopman spectral methods or not?
Thank you so much for all your videos, your book and videos have been the basis of my data-driven dynamics/ machine learning themed bachelor's dissertation. How did people survive before youtube?
21:28 Maybe I am misunderstanding, but I feel that this fluid dynamics example is extremely misleading. The Koopman operator theory is supposed to describe the linear evolution operator for measures over finite dimensional ODEs. The eigenfunctions of the Koopman operators are also the eigenfunction for the Focker-Plank (with no diffusion term) that describes the dynamics of the non-linear ODE. So, the magic sauce here is removing non-linearity at the price of infinitely increasing the dimension. In this fluids example, you are not decreasing the non-linearity by increasing the dimension. You are just approximating the flow as is to be linear. Fundamentally, A^t X still can only linear depend on the initial flow velocity. In addition, there should actually be a Koopman theory for fluid flows. It would describe the evolution operator on measures over the function space of flow fields (in physics we call this Euclidean Field Theory (with no diffusion)).
Absolutely love your content Steve! Is there a relationship between the Koopman operator and Noether's theorem(s) or the Euler-Lagrange equations, or facts about either you can exploit to control or learn dynamical systems?
I really like these amazing videos, but I have a question. Here we are interested in a Koopman invariant subspace; however, the existence of such invariant subspace might not be even possible in general Hilbert space (as the invariant subspace problem is still an open question unless some restriction is applied). What would you think about this?
Amazing content and I still can't believe I find such high quality overview for free. Thank you for all the effort you put into making these videos!
You are very welcome!
Amazing overview! Thank you for all your effort in putting Koopman's theory in a nice, easily-digestible fashion! Keep you your amazing work!
Thanks very much!
I have several published papers on the so-called Koopman-von Neumann Hilbert space representation of Hamiltonian mechanics, yet I know very little about the Koopman operator theory. I feel like there is a language barrier that makes it hard for me to study this topic, so I'm glad that these videos exist.
I'm glad you find them helpful!
Thanks for including references, I read these after your presentation. Well done. Thanks
Thanks, Steve! I am doing research over resolvent mode decomposition, the paper of Ati Sharma was helpful.
Glad it was helpful!
One relatively minor point from the control theory point of view I’d say regardless of nonlinearity if the control scheme is convex, then we will be able to implement a nice controller for the most part. The big challenge is if the model predictive control scheme has to solve a non-convex optimization problem to find even a good control input. If we are in the non-convex regime we don’t have good algorithms that can operate without babysitting, so it becomes untenable for many control applications.
Therefore, non-convexity is the challenge rather than non-linearity.
Very good explained , thank you for this brilliant introduction.
Glad you enjoyed it!
This is a great presentation. I learned a lot from the presentation. Thanks!
Glad you liked it!
Thank you for the great talk! Minor issue: I believe there's a typo at @15:47 : phi_lambda should be x_2, and phi_2mu should be x_2 + (lambda - 2mu)/lambda * x_1^2
At about 23:11, you mention a pseudoinverse that you get by regression. Is an inverse covariance matrix used, and if so, does it follow from the data? Excellent intro and I look forward to the other lectures.
At about 20:52 you derive \phi(x)=e^{-1/x}. You can just as easily show that e^{-\lambda / x} has eigenvalue \lambda. So there's a continuous spectrum, and 0
One more question. The Koopman operator is the adjoint of the Frobenius Perron operator, the latter advancing a density and the former advancing a scalar (an observable.) Is it possible to repeat all the Koopmanism stiuff with the F-P operator? Also, in the conventional use of adjoint methods, you can either advance the F_P forward in time or the K backward, taking the inner product of the scalar and the density at the initial or the final time. Does any of this appear in Koopman spectral methods or not?
I wish I could have seen this earlier. Thank you!
Thanks!
Thank you so much for all your videos, your book and videos have been the basis of my data-driven dynamics/ machine learning themed bachelor's dissertation. How did people survive before youtube?
is x=-1/t a solution to that differential equation @17:29 ?
yep, but he is looking for eigenfunctions of that operator.
21:28 Maybe I am misunderstanding, but I feel that this fluid dynamics example is extremely misleading. The Koopman operator theory is supposed to describe the linear evolution operator for measures over finite dimensional ODEs. The eigenfunctions of the Koopman operators are also the eigenfunction for the Focker-Plank (with no diffusion term) that describes the dynamics of the non-linear ODE. So, the magic sauce here is removing non-linearity at the price of infinitely increasing the dimension. In this fluids example, you are not decreasing the non-linearity by increasing the dimension. You are just approximating the flow as is to be linear. Fundamentally, A^t X still can only linear depend on the initial flow velocity. In addition, there should actually be a Koopman theory for fluid flows. It would describe the evolution operator on measures over the function space of flow fields (in physics we call this Euclidean Field Theory (with no diffusion)).
Essentially, the Koopman modes in the fluids case are the Koopman eigenmodes for individual particles.
Absolutely love your content Steve! Is there a relationship between the Koopman operator and Noether's theorem(s) or the Euler-Lagrange equations, or facts about either you can exploit to control or learn dynamical systems?
"So it's kinda nasty..." I've never thought of math can be so lively :)
Amazing!.Thank you for the lecture.Hope we will meet some day 😄
Thanks!
Would you be able to post or show how the eigenfunctional was found using the laurent series?
Is there a relation with spectral theory and operator algebras and Koopman operator theory?
I really like these amazing videos, but I have a question. Here we are interested in a Koopman invariant subspace; however, the existence of such invariant subspace might not be even possible in general Hilbert space (as the invariant subspace problem is still an open question unless some restriction is applied). What would you think about this?
you know, after i break the delfie hellman I'm gonna enroll in your class.
❤