The Modern Mathematics of Deep Learning

A product of orbitals

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

Evaluate an orbital at each particle position and multiply the values.

The source uses this product to motivate a more structured wavefunction.

Source formula example

\prod_{i=1}^{n}\varphi_{i}(r_{i})

illustration · This direct product generally does not have the required fermionic antisymmetry.

Symbols

$φᵢ$: Orbital function i.
$rᵢ$: Position of particle i.
$n$: Positive number of particles.

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 orbital value

This example’s data flow

You choose

input.first = 1

Calculated here

Result expression: { orbitalValues, product: orbitalValues.reduce((a, b) => a * b, 1) }

orbitalValues = [input.first, 2, 3] Uses: input.first

Returns

Example result

orbitalValues: array (3 items): [1, 2, 3]
product: 6

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The three orbital evaluations are invented scalar values.

Source: Source expression