The function that returns the \(\mathsf{Etr}\) networks.

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

Etr function

<dl><dt>n</dt>
<dd>number of trapezoids to make. Note this will result in a set of
trapezoids. A natural number.</dd>
<dt>h</dt>
<dd>width of trapezoids. A positive real number.</dd>Note: Upon instantiation with any continuous function this neural
network must be fed with \(n+1\) real numbers representing the values
of the function being approximated at the \(n+1\) meshpoints which are
the legs of the \(n\) trapezoids as stipulated in the input parameter \(n\)..</dl>

Arguments

Etr — Etr

<dl>

<dt>n</dt>
<dd>number of trapezoids to make. Note this will result in a set of
trapezoids. A natural number.</dd>


<dt>h</dt>
<dd>width of trapezoids. A positive real number.

Note: Upon instantiation with any continuous function this neural
network must be fed with \(n+1\) real numbers representing the values
of the function being approximated at the \(n+1\) meshpoints which are
the legs of the \(n\) trapezoids as stipulated in the input parameter \(n\)..</dd>

</dl>

Etr: Etr

Description

Usage

Value

Arguments

References

Examples