The Modern Mathematics of Deep Learning

Random input for a PDE example

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 considers a uniformly distributed input and applies the initial function to a stochastic process started at that input.

The toy uses one dimension, zero drift and diffusion, and φ(z)=z². Thus Sₓ,ₜ=x and Y=X².

The example shows how a random input and process produce a random target.

Source formula example

X\sim\mathcal U([0,1]^d),\quad Y=\varphi(\mathcal S_{X,t})

illustration · The displayed x is a chosen point, not a random sample. The mean 1/3 is the analytic expectation of X² for a continuous uniform X on [0,1]. The toy process is deterministic.

Symbols

$X$: An input uniformly distributed on the unit cube.
$\mathcal{S}_{X,t}$: The stochastic process started at X.
$Y$: The resulting target φ(S).

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.

One input value

This example’s data flow

You choose

input.x = 0.5

Calculated here

Result expression: { x, uniformCdf, initialApproximation, process, target, exactMeanTarget: 1 / 3 }

x = input.x Uses: input.x
uniformCdf = x Uses: x
initialApproximation = x Uses: x
process = x Uses: x
target = process ** 2 Uses: process

Returns

Example result

x: 0.5
uniformCdf: 0.5
initialApproximation: 0.5
process: 0.5
target: 0.25
exactMeanTarget: ≈0.3333

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

For the toy zero-drift, zero-diffusion process, Sₓ,ₜ=x. The initial step of the toy approximation is also x. The continuous uniform CDF at x∈[0,1] is x, and the exact expectation ∫₀¹x²dx is 1/3.

Source: Random input example