Effortless estimation of basins of attraction

Abstract

We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the dynamical system flow. The input to the method is a dynamical system evolution rule and a grid that partitions the state space. No prior knowledge of the number, location, or nature of the attractors is required. The method works for arbitrarily high-dimensional dynamical systems, both discrete and continuous. It also works for stroboscopic maps, Poincaré maps, and projections of high-dimensional dynamics to a lower-dimensional space. The method is accompanied by a performant open-source implementation in the DynamicalSystems.jl library. The performance of the method outclasses the naïve approach of evolving initial conditions until convergence to an attractor, even when excluding the task of first identifying the attractors from the comparison. We showcase the power of our implementation on several scenarios, including interlaced chaotic attractors, high-dimensional state spaces, fractal basin boundaries, and interlaced attracting periodic orbits, among others. The output of our method can be straightforwardly used to calculate concepts, such as basin stability and final state sensitivity.

Fast Bayesian Optimization of Needle-in-a-Haystack Problems using Zooming Memory-Based Initialization

“…Through the combination of these two contributions, the optimum region of a NiaH manifold can be quickly discovered in fewer experiments without pigeonholing into local minima. Thus, the three challenges of optimizing NiaH problems are addressed: (1) the challenge of finding a hypervolume within the manifold that contains the needle-like optimum [28,29,11], (2) the challenge of avoiding pigeonholing into local minima [9,1,30,31], (3) the challenge of the polynomially increasing compute times of BO using a GP surrogate [34,35,5,6,36,37]. We demonstrate the implementation of ZoMBI on a 5D analytical Ackley function, a 5D dataset of materials with Poisson's ratios, and a 5D dataset of thermoelectric materials, all of which exhibit a NiaH problem.…”

“…The ZoMBI algorithm is designed specifically to tackle NiaH problems where the basin of attraction containing a global minimum is narrow [30,31]. In our first experiment we explore the question, how does basin of attraction width affect the ability of ZoMBI to find the global minimum?…”

“…sampled memory points to iteratively zoom in the search bounds of the manifold uniquely on each dimension and (2) implementing two custom acquisition functions, LCB Adaptive and EI Abrupt, that actively learn information about the manifold during optimization to tune the sampling of new experimental conditions from a surrogate. The main contributions of this algorithm solve three fundamental challenges of optimizing non-convex Needle-in-a-Haystack problems: (1) the challenge of locating the hypervolume region of the manifold containing the narrow global optimum basin of attraction [28,29,11] is alleviated by introducing iterative search bounds based on learned knowledge of the manifold; (2) unwanted pigeonholing into local minima [30,31,5,6] is avoided by both the zooming mechanics of ZoMBI as well as using the two acquisition functions developed in his paper, LCB Adaptive and EI Abrupt, that tune their hyperparameters through active learning;…”

“…This challenge requires the development of an algorithm that can more quickly determine the plausible region of the manifold where the needle exists. The second challenge for algorithms, such as BO, to optimize NiaH manifolds is in the nature of the acquisition function to pigeonhole sampling into local minima because of the narrowness of the needle's basin of attraction [30,31]. Standard BO acquisition functions, including expected improvement (EI) [32] and lower confidence bound (LCB) [7,12], are static sampling techniques that only adjust sampling based on the output of the surrogate model, which enacts smoothing of the needle [11,5,6].…”

Unpredictability and basin entropy

“…Another natural development of the basin entropy is related to the Wada property. Actually, the detection and characterization of Wada basins has drawn much attention All figures have been computed with Julia programming language using an automatic algorithm [39]. The code for reproducing the plots is available at [19].…”