A function that returns the \(\mathsf{Prd}\) neural networks that
approximates the product of two real numbers when given an appropriate
\(q\), \(\varepsilon\), a real number \(x\) and instantiation with ReLU.
activation.

Do algebraic operations on neural networks. We seek here to implement
in R, operations on neural networks and their resulting approximations. Our operations derive
their descriptions mainly from
Rafi S., Padgett, J.L., and Nakarmi, U. (2024), "Towards an Algebraic Framework For Approximating Functions Using Neural Network Polynomials", <doi:10.48550/arXiv.2402.01058>,
Grohs P., Hornung, F., Jentzen, A. et al. (2023), "Space-time error estimates for deep neural network approximations for differential equations", <doi:10.1007/s10444-022-09970-2>,
Jentzen A., Kuckuck B., von Wurstemberger, P. (2023), "Mathematical Introduction to Deep Learning Methods, Implementations, and Theory" <doi:10.48550/arXiv.2310.20360>.
Our implementation is meant mainly as a pedagogical tool, and proof of concept. Faster implementations with
deeper vectorizations may be made in future versions.

Shakil Rafi

Neural Networks Made Algebraic

Joshua Lee Padgett

Ukash Nakarmi

Prd function

<dl><dt>q</dt>
<dd>a real number in \((2,\infty)\). Accuracy as well as computation
time increases as \(q\) gets closer to \(2\) increases</dd>
<dt>eps</dt>
<dd>a real number in \((0,\infty)\). ccuracy as well as computation
time increases as \(\varepsilon\) gets closer to \(0\) increases</dd></dl>

Arguments

Prd — Prd

<dl>

<dt>q</dt>
<dd>a real number in \((2,\infty)\). Accuracy as well as computation
time increases as \(q\) gets closer to \(2\) increases</dd>


<dt>eps</dt>
<dd>a real number in \((0,\infty)\). ccuracy as well as computation
time increases as \(\varepsilon\) gets closer to \(0\) increases</dd>

</dl>

Prd: Prd

Description

Usage

Value

Arguments

References

Examples