Very interesting for the novice and very nicely presented. I can't think of a better introduction to the topic, and I can't think of a more charismatic and didactical presentation. Thank you 10000 times for sharing this. On the other hand, I believe that there's room for improvement in several parts that are quite incorrect and very superficial. At least, one could give a hint to the possible problems that occur when using the algorithm in real life applications. For example, it is not true what stated at 29:27 that Atilde = U^* A U is a similarity transform because U is not a square matrix. A tilde and A do not generally share the same eigenvalues (not even the dominant ones). Moreover, trying to propagate a linear system in a reduced space could be a very badly conditioned problem. What happens if the SVD basis selects only 2 modes (r=2) and each of these evolves with more than 2 frequencies? (very possible, since the SVD gives no constraints to the frequency content of its structures) Well, the algorithm will not move past step 2.
Correct me if I'm wrong but isn't the point of reducing the dimensionality was to only consider the non-singular subspace of the system? That would mean the system would actually be well-conditioned after the dimensionality reduction step as it is not singular.
At 17:58 the matrix A is defined to be both the rate of change at state x as well as the mapping from one timestep to the next. Given the first definition x2=x1+dt*Ax1 (eulers method for timestepping), while for the latter you would have x2=Ax1.
The later `A' could be I+hA (h=\delta t) for Euler integration or (I-hA/2)^{-1}(I+hA/2) for implicit midpoint, or whatever. But if we find the later A, do we really need to find the original (ODE) A?
35:36 What if somebody jumps into the flow you're measuring thus destroying a house of cards you've decided to be a sound depiction of physical world? You cannot predict when the fellow conscience being decides to have fun diving into the hotel pool.
The fundamental assumption is that we want a linearization that works "well" at every point in the state space. Why does that even make sense to assume? For some arbitrary nonlinear ODE, linearization will be very different at different points in the domain.
There are very few global statements to be made about non linear differential DEs and require assumptions to be made by the designer. However, we generally only want to know local behaviour of the system as phenomena are commonly constraint to a subset of the domain space. Furthermore, prediction tasks can be modeled effectively linearly given the system is sufficiently stable.
The mode is the eigenfunctions that you get from the system. The idea is that this method transforms the nonlinear dynamical system into a linear dynamical system, and the eigenfunctions you get are "modes" for the nonlinear dynamics.
You're a legend. Watched explanations from other professors and yours is by far the easiest to follow. Thanks, Prof. Kutz!
the matthew mcconaughey of dynamical systems. amazing
They both have simular vocabulary for miscellaneous words and expressions on top of similar intonations.
lol😂，now i c why 'magic mike' is also on my youtube recommendation.
got to this place by videos of Steven Bruntons. Hereby, I realize how smart you guys are! Salute, that's not teaching, but art of teaching. Respect!
Man, I found this by accident, but OMG u r the best. AC/DC reference was perfect.
DMD, and SINDy, is amazing stuff. I was exposed to SINDy through a schoolmate and have just found this entire subject extremely fascinating.
Very interesting for the novice and very nicely presented. I can't think of a better introduction to the topic, and I can't think of a more charismatic and didactical presentation. Thank you 10000 times for sharing this.
On the other hand, I believe that there's room for improvement in several parts that are quite incorrect and very superficial. At least, one could give a hint to the possible problems that occur when using the algorithm in real life applications.
For example, it is not true what stated at 29:27 that Atilde = U^* A U is a similarity transform because U is not a square matrix. A tilde and A do not generally share the same eigenvalues (not even the dominant ones).
Moreover, trying to propagate a linear system in a reduced space could be a very badly conditioned problem. What happens if the SVD basis selects only 2 modes (r=2) and each of these evolves with more than 2 frequencies? (very possible, since the SVD gives no constraints to the frequency content of its structures) Well, the algorithm will not move past step 2.
Correct me if I'm wrong but isn't the point of reducing the dimensionality was to only consider the non-singular subspace of the system? That would mean the system would actually be well-conditioned after the dimensionality reduction step as it is not singular.
Absolutely fantastic!! This is truly fantastic and a treasure. Thank You!!
At 17:58 the matrix A is defined to be both the rate of change at state x as well as the mapping from one timestep to the next. Given the first definition x2=x1+dt*Ax1 (eulers method for timestepping), while for the latter you would have x2=Ax1.
The later `A' could be I+hA (h=\delta t) for Euler integration or (I-hA/2)^{-1}(I+hA/2) for implicit midpoint, or whatever. But if we find the later A, do we really need to find the original (ODE) A?
Your teaching is inspirational Nathan, and the material is top notch, as always! Thank you so much
WOW. I wish I had teachers like you.
Professor Nathan, so lovely and enjoying your class.
I really enjoy watching this lecture
This is a great lecture 👌
Amazing lecture, asolutely enjoyed it.
I'm in love with you and your lecture 😍😅
Excellent lecture, thank you for sharing
its such a pleasure watching this !!!!! LOVE THE PRESENTATION
At 33:28, shouldn't he be using U_r for the similarity transform so that it reduces the number of dimensions down to r?
Yes. After he wrote U_r, V_r and Sigma_r on the board, it should be assumed that they are all subscript r in the rest of the discussion.
35:36 What if somebody jumps into the flow you're measuring thus destroying a house of cards you've decided to be a sound depiction of physical world? You cannot predict when the fellow conscience being decides to have fun diving into the hotel pool.
where is the link for next lecture on "sparse identification of nonlinear dynamics" ?
Excellent.
I remember the AC/DC song Dynamite Mode Decomposition. It is quite catchy.
I've never seen a math teacher talking about the AC/DC during the lecture... XD
Can't agree more
I've had a physics techer talk about it... Get it?
22:55 Dirac delta function.
What course was this covered in at Washington?
Amazing. Thank you
very interesting subject.
You almost gave me a heart attack! It was Malcolm who passed.
Hi, Can dmd also work for low dimensional non linear systems. Say for example 10 measurements per snapshot?
Sometimes. If
Lg Clear structural unnderstand lection !
Legend 👏
Is it just me or the professor looks like Matthew McConaughey?
The fundamental assumption is that we want a linearization that works "well" at every point in the state space. Why does that even make sense to assume? For some arbitrary nonlinear ODE, linearization will be very different at different points in the domain.
It's just an approximation. A regression.
There are very few global statements to be made about non linear differential DEs and require assumptions to be made by the designer. However, we generally only want to know local behaviour of the system as phenomena are commonly constraint to a subset of the domain space. Furthermore, prediction tasks can be modeled effectively linearly given the system is sufficiently stable.
What is meant by mode? Why is it named as DYNAMIC MODE DECOMPOSITION?
The mode is the eigenfunctions that you get from the system. The idea is that this method transforms the nonlinear dynamical system into a linear dynamical system, and the eigenfunctions you get are "modes" for the nonlinear dynamics.
This guy looks like Matthew Mcgonagall from Interstellar movie
Low rank structure... Or a low dimensional manifold
Make Things Linear Again