Teaching a Neural Network the Mandelbrot Set

A neural network can be trained to approximate the Mandelbrot set by framing it as a regression problem to predict a smooth, continuous escape-time value. A standard deep residual multi-layer perceptron (MLP) struggles with this task, producing a blurry image that fails to capture the set's intricate details. This failure is attributed to the "spectral bias" of neural networks, which makes them predisposed to learning low-frequency functions and unable to represent fine, high-frequency patterns. The solution presented is to use Gaussian Fourier Features, which transform the input coordinates into a frequency space before feeding them to the network. This technique overcomes spectral bias, enabling the model to learn the high-frequency details and render a sharp, accurate representation of the fractal boundary.