Discretized boundary-oriented progressive learning method for predicting global basins of attraction with few data

Basins of attraction (BoAs) are crucial for evaluating quality of a response and unraveling reliability of complex systems and mechanism of nonlinear phenomena. As a global strategy, however, it will pose a significant challenge to quantify a high-dimensional BoAs due to the curse of dimensionality and the insufficiency of data. This paper proposes a boundary-oriented progressive learning method based on the state space discretization, which aims to perform the classification of dynamics using few samples needed for learning while still achieving high efficiency and accuracy. Using pattern recognition network, training samples are purposefully extracted from the discretized and limited region that covers cells of boundary, disregarding the region outside of it. The region is then refined and identified iteratively to enhance discriminability of data model. This method does not seek to approximate the structure of boundary by refine cells, but rather regards cells as a framework of training neural network. The three typical examples are illustrated to show the power of the proposed method. The results demonstrate that the higher the dimensions, the better cost-effectiveness when compared to state-of-the-art approaches. The performance is even improved by more than 4 orders of magnitude on the computing loads when coping with a formidable six-dimensional BoA with satisfactory accuracy. Also, we discuss how the boundary-oriented progressive learning can improve the overall accuracy and robustness of the data model. Furthermore, this idea has the potential to efficiently handle other tasks of classification of dynamics beyond BoA, from a perspective of engineering.