Dynamical System Parameter Path Optimization using Persistent Homology

TL;DR

The paper introduces a topology-informed framework for navigating high-dimensional dynamical system parameter spaces by leveraging differentiable persistent homology to encode desired dynamical behaviors as topological losses. It couples a parameter-to-state map with a differentiable persistence pipeline and adjoint sensitivity to enable gradient-descent navigation toward parameter regions that realize target topological features such as periodicity, fixed points, or controlled chaos. A structured loss-function library translates persistence diagram traits into concrete dynamical objectives, and the approach is validated on Rössler, magnetic pendulum, and Lorenz systems, demonstrating transitions between chaotic, periodic, and fixed-point regimes. This topology-driven parameter search offers a principled, data-driven mechanism for safe and efficient exploration of complex dynamical responses with potential broad impact in engineering and applied sciences, supported by open-source code.

Abstract

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.

Figures (24)

• Figure 1: Point cloud persistent homology example. (a-h) show the Vietoris-Rips filtration for increasing values of the connectivity parameter and (i) shows the full persistence diagram with the 3 prominent loops labeled in the persistence diagram and the filtration.
• Figure 2: Persistence optimization example. The initial point cloud is shown in (a) and the updated point cloud and persistence diagrams are shown at different points in the optimization process in (b-f). The loss is plotted in (g) with the different points labeled at the respective epoch numbers.
• Figure 3: Diagram demonstrating the map from the parameter space to the loss function as solving the inverse problem from taking a step against the gradient of the loss function to reach a new set of parameters in the parameter space propagated through the persistence diagram and the state space point cloud.
• Figure 4: Example response criteria mapped into persistence diagrams. (Left) Maximizing maximum persistence to promote a large loop in the state space, (Middle) Limiting persistent loops to be close to the diagonal to encourage fixed point behavior, and (Right) Using high persistent entropy to classify chaotic regions in the persistence diagram.
• Figure 5: Lorenz system optimal parameter space paths using the global and local updating schemes. Corresponding persistence diagrams are shown at three points to demonstrate the topological differences between dynamic states.