The Modern Mathematics of Deep Learning

Particle and nucleus coordinates

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Quantum models · Describe wavefunctions, particle coordinates and energy relations.

PDE and quantum models are not operations on these penguin measurements.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 8.2 Equations as guides

A particle configuration records three coordinates for every electron.

The Hamiltonian acts on a wavefunction of the full configuration.

Source formula example

r=\begin{bmatrix}r_{1},\dots,r_{n}\end{bmatrix}\in{\mathbb{R}}^{3\times n}

illustration · This is a coordinate representation, not an energy calculation.

Symbols

$r$: 3 by n matrix of electron positions.
$R$: 3 by p matrix of nuclear positions.
$n$: Number of electrons.
$p$: Number of nuclei.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

First electron x coordinate

This example’s data flow

You choose

input.first = 1

Initial values in code

R = [ [0, 4], [0, 0], [0, 0], ]

Calculated here

Result expression: { electronCoordinates: r, nucleusCoordinates: R, electronCount: r[0].length, nucleusCount: R[0].length, }

r = [ [input.first, 2], [0, 0], [0, 0], ] Uses: input.first

Returns

Example result

electronCoordinates: array (3 items): [array (2 items): [1, 2], array (2 items): [0, 0], array (2 items): [0, 0]]
nucleusCoordinates: array (3 items): [array (2 items): [0, 4], array (2 items): [0, 0], array (2 items): [0, 0]]
electronCount: 2
nucleusCount: 2

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The toy arranges each particle position as a column and uses two electrons and two nuclei.

Source: Source expression