用拓扑分析优化动态系统参数路径，让系统自动走向期望状态。
Dynamical System Parameter Path Optimization using Persistent Homology
- 通过可微的持久性图定义拓扑语言，指导参数调整方向。
- 利用梯度下降在高维参数空间中寻找目标拓扑特征的最优路径。
- 适用于需精准调控系统行为的复杂动力学场景，如生物模型或工程系统。
非线性动力系统复杂难解，通常只有简单系统可解析研究。实际应用中，系统由可调参数定义，参数变化会引发系统响应的显著拓扑变化（分岔）。在高维参数空间中，难以确定如何调整参数以实现期望的系统状态。本文提出一种基于拓扑数据分析的新方法，利用持久性图的可微性，构建直观的拓扑语言，用于促进或抑制系统状态空间中的特定拓扑特征，并通过梯度下降实现参数空间中的最优导航。最终得到一条从初始点到目标参数的路径，使系统响应具备由损失函数定义的期望拓扑特征。我们通过多个动力系统与场景验证了该方法的有效性，展示了如何引导不同拓扑特征的生成及超参数选择对结果的影响。
原文摘要 · Abstract (English)
Nonlinear dynamical systems are complex and typically only simple systems can be analytically studied. In applications, these systems are usually defined with a set of tunable parameters and as the parameters are varied the system response undergoes significant topological changes or bifurcations. In a high dimensional parameter space, it is difficult to determine which direction to vary the system parameters to achieve a desired system response or state. In this paper, we introduce a new approach for optimally navigating a dynamical system parameter space that is rooted in topological data analysis. Specifically we use the differentiability of persistence diagrams to define a topological language for intuitively promoting or deterring different topological features in the state space response of a dynamical system and use gradient descent to optimally move from one point in the parameter space to another. The end result is a path in this space that guides the system to a set of parameters that yield the desired topological features defined by the loss function. We show a number of examples by applying the methods to different dynamical systems and scenarios to demonstrate how to promote different features and how to choose the hyperparameters to achieve different outcomes.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。