The Modern Mathematics of Deep Learning

Wavefunction input and output

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 wavefunction takes all electron coordinates and returns a scalar value.

This calculation explains the corresponding source expression.

Source formula example

\psi\colon\mathbb{R}^{3\times n}\to\mathbb{R}

illustration · This invented wavefunction is not necessarily antisymmetric and does not solve the Hamiltonian eigenproblem.

Symbols

$ψ$: Real-valued wavefunction on the configuration space.
$r$: 3 by n real matrix of electron coordinates.
$n$: Number of electrons.

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

Calculated here

Result expression: { configuration: r, rows: r.length, electrons: r[0].length, psi }

r = [ [input.first, 2], [0, 0], [0, 0], ] Uses: input.first
squaredLength = r.reduce((s, row) => s + row.reduce((a, v) => a + v * v, 0), 0) Uses: r
psi = Math.exp(-squaredLength) Uses: squaredLength

Returns

Example result

configuration: array (3 items): [array (2 items): [1, 2], array (2 items): [0, 0], array (2 items): [0, 0]]
rows: 3
electrons: 2
psi: ≈0.006738

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy is ψ(r)=exp(−Σᵢ‖rᵢ‖²) with n=2. It evaluates one real-valued wavefunction on one configuration.

Source: Source expression