A model-free sampling method for basins of attraction using hybrid active learning (HAL)

Abstract

Understanding basins of attraction (BoA) is often a paramount consideration for nonlinear systems. Most existing approaches for determining high-resolution BoA require prior knowledge of the system’s underlying math model (e.g., differential equation or point mapping for continuous systems, cell mapping for discrete systems, etc.). These approaches, however, become impractical when a system’s dynamics cannot be derived from first principles (e.g., modeling biological systems), or are approximate. This paper introduces a model-free sampling method to obtain BoA. The proposed method is based upon hybrid active learning (HAL) and is designed to find and label the “informative” samples, which efficiently determine the boundary for the BoA. The approach consists of three primary parts: (1) additional sampling on trajectories (AST) to maximize the number of samples obtained from each simulation or experiment; (2) an active learning (AL) algorithm to exploit the local BoA boundary; and (3) a density-based sampling (DBS) method to explore the global BoA boundary. An example of estimating the BoA for a bistable nonlinear system is presented to demonstrate the efficacy of the HAL sampling method.