nnR (version 0.1.0)

inst: inst

Description

The function that instantiates a neural network as created by create_nn().

Usage

inst(neural_network, activation_function, x)

Value

The output of the continuous function that is the instantiation of the given neural network with the given activation function at the given $x$. Where $x$

is of vector size equal to the input layer of the neural network.

Arguments

neural_network

An ordered list of lists, of the type generated by create_nn() where each element in the list of lists is a pair $(W,b)$ representing the weights and biases of that layer.

NOTE: We will call istantiation what Grohs et. al. call "realization".

activation_function

A continuous function applied to the output of each layer. For now we only have ReLU, Sigmoid, and Tanh. Note, all proofs are only valid for ReLU activation.

x

our input to the continuous function formed from activation. Our input will be an element in ${\mathbb{R}}^d$ for some appropriate $d$.

References

Grohs, P., Hornung, F., Jentzen, A. et al. Space-time error estimates for deep neural network approximations for differential equations. (2019). https://arxiv.org/abs/1908.03833.

Definition 1.3.4. Jentzen, A., Kuckuck, B., and von Wurstemberger, P. (2023). Mathematical introduction to deep learning: Methods, implementations, and theory. https://arxiv.org/abs/2310.20360

Very precisely we will use the definition in:

Definition 2.3 in Rafi S., Padgett, J.L., Nakarmi, U. (2024) Towards an Algebraic Framework For Approximating Functions Using Neural Network Polynomials https://arxiv.org/abs/2402.01058

Examples

create_nn(c(1, 3, 5, 6)) |> inst(ReLU, 5)
create_nn(c(3, 3, 5, 6)) |> inst(ReLU, c(4, 4, 4))