The Modern Mathematics of Deep Learning

Continuous residual evolution

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.

Residual networks · Describe residual layer updates and their continuous analogue.

Convolution, recurrent and residual architectures are not part of this linear penguin model.

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: 6.2 Shortcuts between layers

The equation specifies the instantaneous rate of change and initial value. For h(t, φ) = φ, the solution from x is x times eᵗ.

This gives a continuous counterpart to an incremental residual update.

Source formula example

\dot{\phi}(t)=h(t,\phi(t)),\qquad\phi(0)=x,

exact · This example solves one invented scalar differential equation, not an arbitrary residual network.

Symbols

$φ(t)$: State at real time t.
$φ̇(t)$: Time derivative of the state.
$h$: Function defining the rate of change.
$x$: Initial state.

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 stateTime

This example’s data flow

You choose

input.x = 1
input.t = 1

Calculated here

Result expression: { initial: input.x, value, derivative: value, h: value }

value = input.x * Math.exp(input.t) Uses: input.x, input.t

Returns

Example result

initial: 1
value: ≈2.718
derivative: ≈2.718
h: ≈2.718

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The chosen h is the identity and the exact solution is φ(t) = x exp(t).

Source: Source expression