The Modern Mathematics of Deep Learning

A deterministic process in a PDE representation

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.

PDE and stochastic processes · Describe differential equations and stochastic representations.

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: 4.3 Structure from an equation

The source process changes according to a drift μ and a Brownian term scaled by σ. Its initial value is x.

In the special case μ = 0 and σ = 0, the process remains at x. With φ(z) = z², the expectation is x².

A stochastic-process representation can express a PDE solution as an expectation.

Source formula example

d\mathcal{S}_{x,t}=\mu(\mathcal{S}_{x,t})dt+\sigma(\mathcal{S}_{x,t})dB_{t},\quad\mathcal{S}_{x,0}=x,

illustration · Zero drift and diffusion give a deterministic special case. This does not simulate Brownian motion or prove the stochastic representation.

Symbols

$\mathcal{S}_{x,t}$: The source process with initial position x at time zero.
$B_t$: A d-dimensional Brownian motion.
$μ,σ$: Drift and diffusion coefficients.
$φ$: Initial function.

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.

Initial positionTime

This example’s data flow

You choose

input.t = 1
input.x = 2

Calculated here

Result expression: { initialValue, processValue, firstSample: processValue, phi, expectation: phi }

processValue = input.x Uses: input.x
initialValue = input.x Uses: input.x
phi = processValue ** 2 Uses: processValue

Returns

Example result

initialValue: 2
processValue: 2
firstSample: 2
phi: 4
expectation: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source assumes continuous coefficients with suitable growth conditions and describes the stochastic representation for the Kolmogorov PDE. In this special case both coefficient functions vanish; φ(z)=z² and g(x,t)=x². Time is nonnegative.

Source: Source expression