The Modern Mathematics of Deep Learning

Finite Rayleigh quotient and local energy

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

For a finite real state, weight each local energy by the squared state value divided by the total squared length.

These expressions specify the inverse or quantum model.

Source formula example

\frac{\int_{\mathbb{R}^{3\times n}}\overline{\psi(r)}(\mathcal{H}_{R}\psi)(r)\,\mathrm{d}r}{\|\psi\|_{L^{2}}^{2}}={\mathbbm{E}}\left[\frac{(\mathcal{H}_{R}\psi)(X)}{\psi(X)}\right]

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

Symbols

$ψ$: Nonzero square-integrable wavefunction in the operator domain.
$Hᴿ$: Hamiltonian with fixed nuclei.
$X$: Random configuration distributed with density |ψ|²/‖ψ‖².
$ψ̄$: Complex conjugate of ψ.
$\Vert ψ\Vert ²$: Integral of |ψ|².

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.

Second state value

This example’s data flow

You choose

input.second = 1

Calculated here

Result expression: { probabilities, localEnergy, expectedEnergy: probabilities.reduce((sum, p, i) => sum + p * localEnergy[i], 0), quotient: psi.reduce((sum, v, i) => sum + v * Hpsi[i], 0) / normSquared, }

psi = [1, input.second] Uses: input.second
Hpsi = [2 * psi[0], 4 * psi[1]] Uses: psi
normSquared = psi.reduce((sum, v) => sum + v * v, 0) Uses: psi
probabilities = psi.map((v) => (v * v) / normSquared) Uses: psi, normSquared
localEnergy = Hpsi.map((v, i) => v / psi[i]) Uses: Hpsi, psi

Returns

Example result

probabilities: array (2 items): [0.5, 0.5]
localEnergy: array (2 items): [2, 4]
expectedEnergy: 3
quotient: 3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This is an exactly enumerated finite distribution for a real nonzero state and diagonal H. The source uses a normalized |ψ|² probability density and a continuous Hamiltonian. Local energy requires ψ ≠ 0 at the evaluated point.

Source: Source expression