Uploaded June 2024 | Updated September 2026, 3 weeks ago
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A New Species
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![Waffles And Harmonic Motion [Part 1]](https://i.ytimg.com/vi/nKavBl1EWiI/mqdefault.jpg "Waffles And Harmonic Motion [Part 1]
In this two part series, we dig into understanding and quantifying simple harmonic motion (SHM). We try to figure our why oscillating systems that move the way they do, and what ideas from physics govern this motion.
I have unapologetically stolen much of my approach from Richard Feynmans wonderful physics lectures: http://www.feynmanlectures.caltech.edu/I_toc.html. The approach presented here was largely borrowed from Feynman chapters 9 and 21. I encourage anyone interesting in going to deeper to read Feynmans lectures.
Supporting code: http://nbviewer.ipython.org/github/stephencwelch/Acoustics-To-Deep-Learning/blob/master/Waffles%20and%20Harmonic%20Motion%20%5BPart%20I%5D.ipynb
It can be forked on github: https://github.com/stephencwelch/Acoustics-To-Deep-Learning
For more on this, see www.welchlabs.com/blog
@stephencwelch")
![Neural Networks Demystified [Part 5: Numerical Gradient Checking]](https://i.ytimg.com/vi/pHMzNW8Agq4/mqdefault.jpg "Neural Networks Demystified [Part 5: Numerical Gradient Checking]
When building complex systems like neural networks, checking portions of your work can save hours of headache. Here well check our gradient computations.
Supporting code:
https://github.com/stephencwelch/Neural-Networks-Demystified
Link to excellent Stanford tutorial: http://ufldl.stanford.edu/wiki/index.php/UFLDL_Tutorial
In this series, we will build and train a complete Artificial Neural Network in python. New videos every other friday.
Part 1: Data + Architecture
Part 2: Forward Propagation
Part 3: Gradient Descent
Part 4: Backpropagation
Part 5: Numerical Gradient Checking
Part 6: Training
Part 7: Overfitting, Testing, and Regularization
@stephencwelch")
![Imaginary Numbers Are Real [Part 10: Complex Functions]](https://i.ytimg.com/vi/pNp8Qf20-sA/mqdefault.jpg "Imaginary Numbers Are Real [Part 10: Complex Functions]
Supporting Code: https://github.com/stephencwelch/Imaginary-Numbers-Are-Real
More information and resources: http://www.welchlabs.com
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardans Problem
Part 4: Bombellis Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemanns Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: http://www.welchlabs.com/resources.")
![Why Deep Learning Works Unreasonably Well [How Models Learn Part 3]](https://i.ytimg.com/vi/qx7hirqgfuU/mqdefault.jpg "Why Deep Learning Works Unreasonably Well [How Models Learn Part 3]
Take your personal data back with Incogni! Use code WELCHLABS and get 60% off an annual plan: http://incogni.com/welchlabs
Welch Labs Guide to AI: https://www.welchlabs.com/resources/ai-book-ezrzm-msrmc
New Patreon Rewards 33:31- own a piece of Welch Labs history!
https://www.patreon.com/welchlabs
Books & Posters
https://www.welchlabs.com/resources
Sections
0:00 - Intro
4:49 - How Incogni Saves Me Time
6:32 - Part 2 Recap
8:10 - Moving to Two Layers
9:15 - How Activation Functions Fold Space
11:45 - Numerical Walkthrough
13:42 - Universal Approximation Theorem
15:45 - The Geometry of Backpropagation
19:52 - The Geometry of Depth
24:27 - Exponentially Better?
30:23 - Neural Networks Demystifed
31:50 - The Time I Quit YouTube
33:31 - New Patreon Rewards!
Special Thanks to Patrons https://www.patreon.com/welchlabs
Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier Soto
References
Simon Prince, Understanding Deep Learning. https://udlbook.github.io/udlbook/
Liang, Shiyu, and Rayadurgam Srikant. Why deep neural networks for function approximation?. arXiv preprint arXiv:1610.04161 (2016).
Hanin, Boris, and David Rolnick. Deep relu networks have surprisingly few activation patterns. *Advances in neural information processing systems* 32 (2019).
Hanin, Boris, and David Rolnick. Complexity of linear regions in deep networks. *International Conference on Machine Learning*. PMLR, 2019.
Fan, Feng-Lei, et al. Deep relu networks have surprisingly simple polytopes. *arXiv preprint arXiv:2305.09145* (2023).
All Code:
https://github.com/stephencwelch/manim_videos
100k neuron wide example training code: https://github.com/stephencwelch/manim_videos/blob/master/_2025/backprop_3/notebooks/Wide%20Training%20Example.ipynb
Code from viewer Hugo Brouwer that achieves 99.5%+ with less than 100 neurons using Fourier Features!
https://github.com/AgntBrwr/baarle-hertog-fourier-features
Code from viewer Nico Waser that uses 100 neurons:
https://github.com/Waser2004/Illustrated Guide_to_AI/tree/main/Chapter%204%20-%20Deep%20Learning
Written by: Stephen Welch
Produced by: Stephen Welch, Sam Baskin, and Pranav Gundu
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![How to Science [Part 5: Mathematics]](https://i.ytimg.com/vi/sTSNTWKGkbw/mqdefault.jpg "How to Science [Part 5: Mathematics]
pdf: http://www.welchlabs.com/guides
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https://www.patreon.com/welchlabs
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