Deep Learning-based Analysis of Basins of Attraction

TL;DR

The paper tackles the computational challenge of characterizing basins of attraction in complex dynamical systems by using convolutional neural networks to predict key unpredictability metrics. Basins are represented as images and fed into CNNs from multiple architectures, with ResNet50 emerging as the most effective in terms of accuracy and speed. The study demonstrates that ResNet50 can predict fractal dimension $FDim$, basin entropy $Sb$, boundary basin entropy $Sbb$, and the Wada property with high fidelity on a large test set while delivering massive speedups over traditional methods. This approach enables scalable, rapid exploration of dynamical behaviors across diverse systems, potentially transforming how basin analysis is conducted in practice.

Abstract

This research addresses the challenge of characterizing the complexity and unpredictability of basins within various dynamical systems. The main focus is on demonstrating the efficiency of convolutional neural networks (CNNs) in this field. Conventional methods become computationally demanding when analyzing multiple basins of attraction across different parameters of dynamical systems. Our research presents an innovative approach that employs CNN architectures for this purpose, showcasing their superior performance in comparison to conventional methods. We conduct a comparative analysis of various CNN models, highlighting the effectiveness of our proposed characterization method while acknowledging the validity of prior approaches. The findings not only showcase the potential of CNNs but also emphasize their significance in advancing the exploration of diverse behaviors within dynamical systems.

Figures (4)

• Figure 1: Examples of basins from the dynamical systems used in our work. (a) This figure corresponds to a basin of attraction from the Duffing oscillator, (b) corresponds to a basin from the Newton fractal, (c) basins of attraction of the forced damped pendulum, (d) is an exit basin from the Hénon-Heiles Hamiltonian system, and finally (e) is an example of a basin of attraction of the magnetic pendulum with $4$ magnets. Further details about these dynamical systems can be found in Appendix \ref{['Appendix']}.
• Figure 2: Schematic representation of the workflow followed in this work. First, we compute basins from various dynamical systems and save their metrics and file paths in a dataframe, which is then stored as a .csv file. This file is utilized later by the CNN to access the basin information. Three distinct sets of images are taken into account: training, validation, and testing, with each computed basin being allocated to one of these image sets. During training, batches of 16 random basins are loaded from the .csv file, reshaped into grayscale images, and fed into the CNN. Four identical CNNs have been trained to accurately predict the desired metrics after training.
• Figure 3: Distribution of values of each metric on all image sets. The blue histogram denotes the training set, the orange denotes the validation image set, and the green denotes the test image set. It can be seen that the histograms do not follow a uniform distribution, which indirectly introduces a bias during training. However, by having the three sets with a similar distribution, the bias is minimized in our results.
• Figure 4: Performance of the ResNet50 architecture at predicting the $\mathbf{14384}$ basins of attraction from the test set. The linear regressions show the comparison between the true values and the predicted values for each metric studied in our work. Each regression is followed in its right side with a histogram where the difference of the true and predicted values is shown. Such histogram approximates to a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$. Since the estimation of the Wada property is a classification task, a more practical representation is the confusion matrix between the predictions and the true values. Such confusion matrix is shown in the bottom right of the image, showing the mean value of the accuracy for the prediction of $1000$ subsets of the test set, each comprising $1000$ random basins, yielding an accuracy of $99.8 \pm 2.76 \%$. proving the efficiency of the ResNet50 architecture.