Kernel Learning for Robust Dynamic Mode Decomposition

[@insightfool]

Thank you for such a clear explanation of this topic!

[@alialedarvish4192]

Thank you for your excellent presentation

[@fly-code]

great job!!!

[@zhihuachen3613]

Great work! 非常棒的研究！

[@NeoxX317]

Great work !!

[@hfkssadfrew]

Very interesting work!

[@AyyappanHabel]

Very interesting work

[@1337RecklessX]

Great work, I am interested in the implication of Kuramoto model of synchronization in neural oscillation and its impact on consciousness.

[@zhenpeng7031]

interesting work. the DMD, SINDy works to unforced rather than the nonlinear system. however, most of the real world system are non-autonomous. How can the LANDO method be applied to a nonlinear system with unknown external excitation.

[@peterj.baddoo3813]

Thanks for the question! There are a couple of ways to model this. One is to incorporate an unknown control variable into the model as we describe in appendix C. For a non-autonomous system you could include time as an explicit function of the kernel. On the other hand, if the transition matrix of the (nonlinear) system is varying in time then you could use the online version of the algorithm (described in appendix B) with an exponential weighting factor or windowing.

[@zhenpeng7031]

@@peterj.baddoo3813 thanks for your valuabe respond. I will follow up on this paper.

[@zhenpeng7031]

@@peterj.baddoo3813 Hi, Peter, Thanks for your reply. I've carefully read appendix C. Is the control force should be pre-known input, like DMDc. My question is the situation of an unknown control force. Hope to hear from you.

[@EtienneADPienaar]

Interesting and excellent presentation! I have two questions: 1) How does it perform for small samples? E.g., when you generate a short trajectory for the Lorenz system? 2) The dynamical systems you've presented are deterministic. How robust is the methodology where the systems are stochastic? E.g., a non-linear system of Stochastic Differential Equations.

[@peterj.baddoo3813]

Great questions! 1) It will depend on your aims but, as with many of these methods, more data is usually better. We find that a quantitative description of the spectrum needs nonlinear transients whereas a qualitative reconstruction doesn't need much data. Of course, the rank of the data is more important than the number of samples, so samples from different nonlinear regimes can be helpful. We are also working on a physics-informed version that requires far fewer samples than usual. 2) I have not tried the method yet for SDEs but I hope to in the future!

[@krishnaaditya2086]

Awesome Thanks!

[@harshavardhans3998]

This looks really interesting. I have been using SINDy to discover the dynamics of my time series data and the results are not that great. I'm curious to apply LANDO and check what could be the difference. However, I have one question, do you think LANDO can capture dynamics if the data is stochastic and are observed at very few timepoints?

[@peterj.baddoo3813]

Thanks for the question, that sounds like a challenging scenario but it could be worth a try with LANDO! Sometimes the kernel representation can uncover a latent space that cannot be represented with finite-dimensional features. This can allow more efficient model identification, which could be relevant in your case.

[@harshavardhans3998]

@@peterj.baddoo3813 Thank you for your answer.

[@kouider76]

Thank you for this great presentation. I will defnitely consider projecting this method to the case of dynamic structure behaviour especialy active vibration control. Do you have the code open access ?

[@peterj.baddoo3813]

Thanks for your comment, Kouider! The code will be published open access here in the coming days: github.com/baddoo/LANDO

[@kouider76]

@@peterj.baddoo3813 Thanks @Peter. Waiting for more videos such this

[@PhDHugo]

I liked the structure of your presentation, how did you edit the video like that? I would like to do the same for some activities at my college.

[@peterj.baddoo3813]

Hi Hugo, this was recorded using a "lightboard studio" e.g. www.lightboard.info/. You can see many great lightboard presentations on Steve Brunton's channel: kzbin.info

[@jonathansaunders7665]

Very interesting stuff and well explained! Just a small question, if a mapping is linear in the both the first and the second arguments, does that make it bilinear?

[@peterj.baddoo3813]

That's a very astute point; the standard linear kernel used in DMD (e.g. 13:08 and 15:30) is bilinear although more generic kernels such as Gaussian and polynomial are not!

[@sebastiangutierrez6424]

Really interesting!! I've two questions. 1) Have you tested this method with equations that have multiple scale phenomenon, like the Navier Stokes equation? 2) Is the method robust under perturbation of the data ? For example, adding to each measurements the realization of a normal distribution.

[@peterj.baddoo3813]

Hi Sebastian, thanks for the questions! 1) We are currently testing the method on data from channel flow simulations to learn the full Navier-Stokes equations! There is scope to include the effects of multiple scales in kernel design. 2) We discuss the sensitivity to noise in appendix E of the arXiv paper (arxiv.org/abs/2106.01510). Some problems might require smoothing the data before applying LANDO (e.g. via total-variation regularised differentiation).

[@sebastiangutierrez6424]

@@peterj.baddoo3813 Thanks a lot for the answers! Your work is really interesting. About the multiscale in kernel design, are multiple scales included by the different magnitudes of the weights for each kernel? I have an additional question, but it's about the general framework of data driven PDE/ODE identification. Do you know if these methods have been applied to delay ODEs?

[@peterj.baddoo3813]

@@sebastiangutierrez6424 Sure, you can include this both through the choice of weights and the type of functions included in the kernel. Similar methods have been applied to delay differential equations, but only in the linear case e.g. www.sciencedirect.com/science/article/pii/S2405896318309832

[@sebastiangutierrez6424]

@@peterj.baddoo3813 Thanks a lot !

[@phillustrator]

RIP man