The Modern Mathematics of Deep Learning

Sampling 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 expresses a PDE solution as an expectation. It then approximates that expectation with a mean of independent sample values.

This toy uses the given list as sample values and φ(s) = s².

Sampling can turn an expectation into a finite computation.

Source formula example

g(x,t)={\mathbbm{E}}[\varphi(\mathcal{S}_{x,t})]\approx\frac{1}{n}\sum_{i=1}^{n}\varphi(\mathcal{S}_{x,t}^{(i)}),

approximation · The fixed toy values are not simulated independent draws. Their mean is not generally the exact expectation and does not prove the PDE result.

Symbols

$φ$: Initial condition in the PDE representation.
$\mathcal{S}_{x,t}$: The source stochastic process at position x and time t.

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.

Third sample

This example’s data flow

You choose

input.last = 2

Calculated here

Result expression: { values, estimate: values.reduce((a, b) => a + b, 0) / values.length }

samples = [0, 1, input.last] Uses: input.last
values = samples.map((s) => s * s) Uses: samples

Returns

Example result

values: array (3 items): [0, 1, 4]
estimate: ≈1.667

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source discusses continuous coefficients and suitable growth conditions, independent identically distributed process samples, and further conditions for uniform approximation. The toy computes only the sample average.

Source: Source expression