The Modern Mathematics of Deep Learning

Drift and diffusion coefficients

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

In the stochastic differential equation, μ is the drift coefficient and σ is the diffusion coefficient.

The coefficients specify deterministic and random changes in the process.

Source formula example

\mu,\sigma

illustration · This is one scalar Euler step with a supplied increment; it does not solve a stochastic differential equation exactly or establish its distribution.

Symbols

$μ$: Drift coefficient function.
$σ$: Diffusion coefficient function.
$Δt$: A positive time step.
$ΔB$: A Brownian increment.

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.

This example’s data flow

Initial values in code

current = 1
drift = 2
diffusion = 0.5
dt = 0.1
brownianIncrement = 0.2

Calculated here

Result expression: { current, drift, diffusion, dt, brownianIncrement, next: current + drift * dt + diffusion * brownianIncrement, }

Returns

Example result

current: 1
drift: 2
diffusion: 0.5
dt: 0.1
brownianIncrement: 0.2
next: 1.3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. This is one scalar Euler step with a supplied increment; it does not solve a stochastic differential equation exactly or establish its distribution.

Source: Inline notation