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Functionals & Functional Derivatives | Calculus of Variations | Visualizations

We can minimize a Functional (Function of a Function) by setting the first Functional Derivative (=Gâteaux Derivative) to zero. Here are the notes: https://raw.githubusercontent.com/Cey... A Function maps a scalar/vector/matrix to a scalar/vector/matrix. We have seen it multiple times, we know how to take derivatives etc. But now imagine something takes in a function and outputs a scalar/vector/matrix? At first this seems more complicated. Situations like these arise for instance in Lagrangian and Hamiltonian Mechanics or when deriving probability density functions from a maximum entropy principle. But a more intuitive example: Say you want to take your car from Berlin to Munich. There are quite a lot of possible routes to take, each with a potentially different velocity and height profile. Now imagine you have a function that associates each point in time over the route with a position on the map. You could use this to deduce the height-and velocity profile. A Functional would now be a function that takes in the route and outputs the fuel consumption, i.e. mapping from a function to a scalar. Then, you might be interested in minimizing your fuel consumption, so you seek the minimum of a Functional. First Derivative equals zero, right? But how do you take the functional derivative. All of this and more will be answered in the video. ;) ------- 📝 : Check out the GitHub Repository of the channel, where I upload all the handwritten notes and source-code files (contributions are very welcome): https://github.com/Ceyron/machine-lea... 📢 : Follow me on LinkedIn or Twitter for updates on the channel and other cool Machine Learning & Simulation stuff:   / felix-koehler   and   / felix_m_koehler   💸 : If you want to support my work on the channel, you can become a Patreon here:   / mlsim   ------- Timestamps: 00:00 Introduction 00:49 Can't we just use Newtonian Mechanics? 01:27 Defining Energies and Parameters 04:21 "Average Difference in Energy" 06:20 A Functional 07:11 Example 1 08:46 Example 2 09:56 Example 3 11:18 Comparing the Examples 12:20 Visualizing the Examples 13:23 Mathematical Definition of a Functional 15:22 Concept of Minimizing a Functional 16:22 Intro to the Functional Derivative 19:43 Example: Minimizing the Functional 22:53 Rearrange for y 25:38 Fundamental Lemma of Calculus of Variations 26:55 Rediscovering Newtonian Mechanics 28:07 Solving the ODE 29:31 Summary: Functional Derivatives 30:35 Outro