Neural Fractals : Generating Fractals Using Complex Valued Neural Networks

Recurrent Neural Networks (RNN) have demonstrated a significant ability of learning complicated dynamics So I wanted to check can we use them to generate fractals?

Mandelbrot Set

A simple however not simplest way to see a fractal is trying to plot the members of the Mandelbrot set in the complex plane. But what is the Mandelbrot set? consider a complex function $f_c(z) = z^2 + c$ then a complex number c is a member of Mandelbrot set if the following sequence does not diverge to infinity.

\[f_c(0), f_c(f_c(0)), f_c(f_c(f_c(0))), ...\]

if you try to draw the members in the complex plane you will see an interesting pattern with amazing self-similarities:

Mandelbrot Set

It’s cool. but what if we use more a sophisticated function $f_c$ ?

A Complex Recurrent Neural Network

Instead of $f_c(z) = z^2 + c$ we can use a more complicated function. for example:

\[g_c(z) = NN_{\theta}(z, c)\]

where $NN_{\theta}$ is a neural network parametrized by a set of parameters $\theta$. Now the sequence

\[g_c(0), g_c(g_c(0)), g_c(g_c(g_c(0))), ...\]

can be expressed using a recurrent neural network. Note since this is gonna be a complex function, the neural network should also be a complex-valued neural network (CVNN). Implementing a network with complex parameters is easy. See this for an example. So I created a complex RNN and tried to plot the associated set. My method was very similar to what explained here.

First I created a complex neural network with random parameters $\theta$
Then applying the network on a set of random numbers sampled uniformly from the complex plane repeatedly for about 50 iterations.
I assigned each sample that remained in a boundary $|c| \le r$ to the corresponding pixel in the resulting image .

I repeated the above procedure until collecting enough samples. After applying a few enhancements to the resulting image, the result was amazing!

Gray Level Fractal