Zhornyak Lyra, Hsieh M Ani, Forgoston Eric
Department of Computer and Information Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
Chaos. 2024 May 1;34(5). doi: 10.1063/5.0204714.
The construction of bifurcation diagrams is an essential component of understanding nonlinear dynamical systems. The task can be challenging when one knows the equations of the dynamical system and becomes much more difficult if only the underlying data associated with the system are available. In this work, we present a transformer-based method to directly estimate the bifurcation diagram using only noisy data associated with an arbitrary dynamical system. By splitting a bifurcation diagram into segments at bifurcation points, the transformer is trained to simultaneously predict how many segments are present and to minimize the loss with respect to the predicted position, shape, and asymptotic stability of each predicted segment. The trained model is shown, both quantitatively and qualitatively, to reliably estimate the structure of the bifurcation diagram for arbitrarily generated one- and two-dimensional systems experiencing a codimension-one bifurcation with as few as 30 trajectories. We show that the method is robust to noise in both the state variable and the system parameter.
分岔图的构建是理解非线性动力系统的一个重要组成部分。当知道动力系统的方程时，这项任务可能具有挑战性，而如果仅能获取与系统相关的基础数据，则难度会大大增加。在这项工作中，我们提出了一种基于Transformer的方法，该方法仅使用与任意动力系统相关的噪声数据来直接估计分岔图。通过在分岔点将分岔图分割成段，训练Transformer以同时预测存在多少段，并使关于每个预测段的预测位置、形状和渐近稳定性的损失最小化。定量和定性结果均表明，训练好的模型能够可靠地估计任意生成的经历一维余维一分岔的一维和二维系统的分岔图结构，所需轨迹少至30条。我们表明，该方法对状态变量和系统参数中的噪声具有鲁棒性。
