The Modern Mathematics of Deep Learning

A one-dimensional 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

A partial derivative describes change along one input while keeping the others fixed. Here time is t and position is x.

Choose σ = 1, μ = 0, and initial function φ(x) = x². Then g(x,t) = x² + t makes both sides equal to 1.

The source uses this kind of PDE to connect random processes and neural-network approximation.

Source formula example

\frac{\partial g}{\partial t}(x,t)=\frac{1}{2}{\operatorname{Tr}}\big(\sigma(x,t)[\sigma(x,t)]^{*}\nabla_{x}^{2}g(x,t)\big)+\langle\mu(x,t),\nabla_{x}g(x,t)\rangle,\quad g(x,0)=\varphi(x),

illustration · This checks one explicit one-dimensional solution with constant coefficients. It is not a general PDE solver and does not prove the approximation results.

Symbols

$g$: A function on ℝᵈ × [0,∞). In the toy d = 1.
$σ$: A d-by-d coefficient matrix; in the toy it is the scalar 1.
$μ$: A d-vector coefficient; in the toy it is zero.
$φ$: The initial condition.

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.

Position xTime t

This example’s data flow

You choose

input.t = 1
input.x = 2

Initial values in code

timeDerivative = 1
secondPositionDerivative = 2

Calculated here

Result expression: { value, timeDerivative, positionDerivative, secondPositionDerivative, rightSide, initialValue: x * x, sigma: 1, mu: 0, }

x = input.x Uses: input.x
t = input.t Uses: input.t
value = x * x + t Uses: x, t
positionDerivative = 2 * x Uses: x
rightSide = 0.5 * secondPositionDerivative Uses: secondPositionDerivative

Returns

Example result

value: 5
timeDerivative: 1
positionDerivative: 4
secondPositionDerivative: 2
rightSide: 1
initialValue: 4
sigma: 1
mu: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source states a linear Kolmogorov PDE with initial value g(x,0) = φ(x), continuous coefficients and suitable growth conditions. The full expression includes a matrix trace and spatial gradient and Hessian; in one dimension these reduce to the operations shown in the toy.

Source: Source expression