Understanding POD: the Proper Orthogonal Decomposition

This was a lot of fun to make! 3blue1brown has inspired me a lot to make a math video with cool animations! This is my take on the Proper Orthogonal Decomposition, an incredibly powerful and useful algorithm for data analysis ! I'm a fluid dynamicist by trade, so my examples come from there. Hope you found this useful!

This was produced with Manim, a python library developed by 3blue1brown and maintained by the community to make mathematical animation videos! Thanks to all contributors for making this possible!

0:00 Intro
0:29 2D Measurements
1:34 Optimal basis vectors
3:14 Basis vectors in 3D
3:55 Higher dimensional data
4:44 Building the data matrix A
5:10 Formal definition of POD
6:34 The spatial mode matrix U
7:18 The energy matrix Sigma
7:45 The temporal mode matrix V
8:06 A simple traveling wave example
10:06 My take on interpretation of POD modes

[ahmeda]

I think this is the best explanation of POD I have ever seen.

[rarelycomments]

Truly exceptional teaching. If only more lecturers had your skills!

[zachzach]

This is exactly what I need right now. Thank you for the amazing content. Please keep doing this!

[TheLarissaCoutinho]

Excellent! Please, continue with this series of videos. Can't describe how much helpful and clear they are!

[SlashSung]

Very good visualization of the interpretation of the V matrix. I just needed it, thank you

[julianroth]

I am currently working on POD-ROM and I must say that I really like your video. Amazing job! I especially like how at the example of the flow around the cylinder you were able to visually explain the essence of the POD. Very nice!

[Soul-rrus]

definitely one of the cooler SVD visualizations I've seen

[ad]

Thanks for keeping the subject understandable, now I can keep this knowledge in my head and recall this days, months, years from now. The animation are excellect.

[tomcross]

Excellent video Fernando - your take on the subject really hasn't got the attention it deserves - amazing work

[umedina]

Amazing video! Thanks for sharing, really insightful. Looking forward for a continuation video were you show how U represents the eigenvectors of the correlation matrix, and the physical interpretation behind them! Still fuzzy on what a mode really is.

Thanks again!

[shivamsundeep]

Thanks a lot....This is the best POD explaination I have seen.... Very helpful... Keep it up.

[tadeumf]

Very instructive video! Clear and concise explanation of a rather abstract mathematical tool that has very interesting applications in the fluid dynamics field.

[dolfinskip]

Amazing explanation... the visualisation helped a lot! Thank you and may your ilk prosper!

[mateuscarvalho]

This is the best video about POD I have seen. Thanks a lot!

[TheRubencho]

What an amazing explanation. Thank you very much!

[chiamatthew]

Great video Fernando. Is there a website that I can view relevant code to the fluid dynamic example you showed? I would love to explore that more. Thanks!

[SKKarthick]

Amazing job in visualizing the math man! kudos!

[lixinrong]

excellent visualization! Very helpful!💖

[brtymn]

Thank you Fernando, very well explained.

[GRAYgauss]

Very well presented, I skipped through a bit, but it truly does feel like an homage to 2B1B and you demonstrate clear understanding. Keep up the good work, I'll be following you. I particularly like that you take your profession and apply it. You add some novelty by having a unique and deep perspective. If I had anything to recommend, it's to not be TOO simple like every other channel and provide abstract/high level knowledge relevant to you/your interests/perspective.