XOR: A Simple Neural Network in Detail

Interactions: The Future of Scientific Communication

For all of human history, mathematical and scientific communication has been carried out using static text and diagrams. Sometime during this century, there will be a complete shift towards dynamic, interactive media. That is, mathematical and scientific communication will be carried out primarily using text and diagrams dancing on the computer screen, interacting all the time with the user. I call these dynamic interactive media “Interactions”. One current way to enable them is using the HTML5 “canvas” element with JavaScript on the web page.

Some of the most initial, primitive, halting steps in this direction are on display on some of today’s websites, and indeed, on this one. Much of science and engineering is enabled using differential equations, and in the future, along with the differential equations, there will be visual interactions with the differential equations, tailored to the application at hand. Likewise for all other types of related mathematics.

The result will be nothing less than the democratization of science. Advanced researchers will be able to communicate their ideas to any person with basic scientific literacy, rather than just their immediate peers. As an outcome of this, there will be a lot more cross-fertilization between different fields of science. The key to realizing this vision in its entirety is to come up with a set of general purpose APIs that can be used across a variety of applications. Along with these APIs, there should be a set of “design patterns” that guide the writers in creating the Interactions.

You may argue, perhaps with some justification, that what I call an Interaction is nothing but a simulation. That may well be true in some cases. But in general, the difference between Interactions and simulations is that while the purpose of simulations is to display a visual solution to an equation, the purpose of Interactions is to communicate the content of an equation in a broader way. The difference between Interactions and Mathematica is that while Mathematica is a system for computation, Interactions are a system for communication which can include computation.

The Interactions presented on this website in the context of the machine learning pages have a light grey background. They are intended to be intuitive to use. For most of the Interactions, you just have to hover over the circles or connecting lines (neurons or weights), with your mouse, or press the appropriate buttons. A brief description of how to engage with each Interaction is given in the paragraph preceding it. I’ll mention that I still use ordinary static diagrams at places – they have a plain white background.

The machine learning Interactions presented on this website are as follows:

Math: Functions

Introducing Neural Networks: Basics

XOR: Playing with the Weights

XOR: Playing with Training

XOR: Backpropagation in Detail

All of these Interactions make use of the HTML5 "canvas" element and are coded in JavaScript. The code for the above Interactions is hosted at GitHub.

An excellent and very simple example of how equations can be made interactive is at the Maxwell’s Equations website. The author achieves the ingenious effect of making every term in each equation, including the equals signs, a clickable hyperlink which leads to a page that describes that term. It is a great site for understanding Maxwell’s equations, which encompass all of electricity and magnetism.

While the Interaction in the above website is specific to Maxwell's equations, the Interactions demonstrated in the physics equations below are perfectly general and can be applied to any mathematical expression or equation whatsoever. They use MathJax, the gold standard for mathematical publishing on the Internet, which is written in JavaScript. If you hover over each term in the equations below, you will see a popup briefly describing that term.

\begin{align} \texttip{F}{Force} = \texttip{m}{Mass} \texttip{a}{Acceleration} \end{align}

\begin{align}
\texttip{\nabla \times}{Curl, the circulation around a loop} \texttip{\vec{\mathbf{B}}}{Magnetic field} -\, \texttip{\frac1c}{One divided by the speed of light}\, \texttip{\frac{\partial\vec{\mathbf{E}}}{\partial t}}{Partial derivative of electric field

with respect to time} & = \texttip{\frac{4\pi}{c}}{4 times pi divided by

the speed of light}\texttip{\vec{\mathbf{j}}}{The electric current} \\
\texttip{\nabla \cdot}{Divergence, the flux through

a closed surface} \texttip{\vec{\mathbf{E}}}{Electric field} & = \texttip{4 \pi}{4 times pi} \texttip{\rho}{The charge inside the surface} \\
\texttip{\nabla \times}{Curl, the circulation around a loop} \texttip{\vec{\mathbf{E}}}{Electric field}\, +\, \texttip{\frac1c}{One divided by the speed of light}\, \texttip{\frac{\partial\vec{\mathbf{B}}}{\partial t}}{Partial derivative of magnetic field

with respect to time} & = \texttip{\vec{\mathbf{0}}}{The zero vector} \\
\texttip{\nabla \cdot}{Divergence, the flux through

a closed surface} \texttip{\vec{\mathbf{B}}}{Magnetic field} & = \texttip{0}{Zero (scalar)}
\end{align}

\begin{align}
\texttip{i}{The imaginary unit which is

the square root of -1} \texttip{\hbar}{The Planck's constant} \texttip{\frac{\partial}{\partial t}\Psi(\mathbf{r},t)}{The partial derivative of the wave function

(which is a function of position and time)

with respect to time} = \texttip{\hat H}{The Hamiltonian operator} \texttip{\Psi(\mathbf{r},t)}{The Hamiltonian operator is applied

to the wave function}
\end{align}

I will point out that this is already a remarkable result. If everyone publishing mathematical content everywhere used this
system, their mathematics would be far more accessible to readers who don't understand the symbols off-hand. You cannot get
such a result on paper or pdf files; only on web pages. This is why I say, in my first paragraph above, that in the future
all mathematical and scientific communication will be via text and diagrams dancing on the computer screen. The popups in the equations above are brief; they can be far more elaborate.

Copyright © 2022 by Sandeep Jain. All rights reserved.