The Modern Mathematics of Deep Learning

Wavefunction and density

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

The squared magnitude of the wavefunction is normalized by its squared L² norm to obtain a probability density.

Normalization makes the total probability equal one.

Source formula example

|\psi(r)|^{2}

illustration · This finite two-position analogue replaces the integral with a sum; it is not a continuous electron-density calculation.

Symbols

$ψ$: The wavefunction.
$r$: The collection of electron positions.
$\Vert ψ\Vert ²L²$: The integral of |ψ|² over positions.

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.

This example’s data flow

Initial values in code

psi = [1, 2]

Calculated here

Result expression: { psi, squared, normSquared, probabilities: squared.map((v) => v / normSquared), }

squared = psi.map((v) => v * v) Uses: psi
normSquared = squared.reduce((a, b) => a + b, 0) Uses: squared

Returns

Example result

psi: array (2 items): [1, 2]
squared: array (2 items): [1, 4]
normSquared: 5
probabilities: array (2 items): [0.2, 0.8]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. This finite two-position analogue replaces the integral with a sum; it is not a continuous electron-density calculation.

Source: Inline notation