The Modern Mathematics of Deep Learning

Eigenvalue equation

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

An eigenvector keeps its direction when a matrix acts on it; its eigenvalue is the scale factor.

These expressions specify the inverse or quantum model.

Source formula example

\mathcal{H}_{R}\psi=\lambda_{\psi}\psi,\hskip 28.45274pt

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$Hᴿ$: Hamiltonian operator for fixed nuclear positions R.
$ψ$: Nonzero wavefunction in the operator domain.
$λψ$: Eigenvalue associated with ψ.

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.

Nonzero amplitude

This example’s data flow

You choose

input.amplitude = 1

Calculated here

Result expression: { psi, Hpsi, lambdaPsi: psi.map((v) => 2 * v) }

psi = [input.amplitude, 0] Uses: input.amplitude
Hpsi = [2 * psi[0], 3 * psi[1]] Uses: psi

Returns

Example result

psi: array (2 items): [1, 0]
Hpsi: array (2 items): [2, 0]
lambdaPsi: array (2 items): [2, 0]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This invented finite diagonal matrix is an analogy for the operator equation; it does not solve a continuous quantum system.

Source: Source expression