# Learning with Mandelbrot and Julia

arXiv: https://arxiv.org/abs/2509.00903 · HTML: https://mlsanity.com/paper/2509.00903/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 31 Aug 2025.

## Builds on (most cited first, 15 of 15)

- [Deep learning for time series classification: a review](https://mlsanity.com/paper/1809.04356/) (2018)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [InceptionTime: Finding AlexNet for Time Series Classification](https://mlsanity.com/paper/1909.04939/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Pre-training without Natural Images](https://mlsanity.com/paper/2101.08515/) (2021)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Improving Fractal Pre-training](https://mlsanity.com/paper/2110.03091/) (2021)
- [Physics guided neural networks for modelling of non-linear dynamics](https://mlsanity.com/paper/2205.06858/) (2022)
- [Classification of chaotic time series with deep learning](https://mlsanity.com/paper/1908.06848/) (2019)
- [Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations](https://mlsanity.com/paper/2407.19707/) (2024)
- [Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices](https://mlsanity.com/paper/2507.09782/) (2025)
- [Learning Fractals by Gradient Descent](https://mlsanity.com/paper/2303.12722/) (2023)

## Abstract (arXiv)

Recent developments in applied mathematics increasingly employ machine learning (ML)-particularly supervised learning-to accelerate numerical computations, such as solving nonlinear partial differential equations. In this work, we extend such techniques to objects of a more theoretical nature: the classification and structural analysis of fractal sets. Focusing on the Mandelbrot and Julia sets as principal examples, we demonstrate that supervised learning methods-including Classification and Regression Trees (CART), K-Nearest Neighbors (KNN), Multilayer Perceptrons (MLP), and Recurrent Neural Networks using both Long Short-Term Memory (LSTM) and Bidirectional LSTM (BiLSTM), Random Forests (RF), and Convolutional Neural Networks (CNN)-can classify fractal points with significantly higher predictive accuracy and substantially lower computational cost than traditional numerical approaches, such as the thresholding technique. These improvements are consistent across a range of models and evaluation metrics. Notably, KNN and RF exhibit the best overall performance, and comparative analyses between models (e.g., KNN vs. LSTM) suggest the presence of novel regularity properties in these mathematical structures. Collectively, our findings indicate that ML not only enhances classification efficiency but also offers promising avenues for generating new insights, intuitions, and conjectures within pure mathematics.