🤖 AI Summary
High-dimensional parameter optimization in nonlinear dynamical systems is hindered by the non-differentiability of topological responses (e.g., attractors, periodic orbits) and the lack of gradient information for structural features.
Method: This paper introduces a topology-guided gradient descent method grounded in the differentiability of persistent homology. It integrates differentiable persistence diagrams into a parametric optimization framework, constructing a differentiable topological loss function. The approach unifies persistent homology, differentiable topological data analysis, and numerical dynamical simulation (e.g., Runge–Kutta), enabling end-to-end, gradient-driven parameter path planning.
Contribution/Results: Evaluated on canonical chaotic and bifurcating systems—including Lorenz, Rössler, and Hindmarsh–Rose—the method achieves precise control over target topological invariants (e.g., number of loops, connected components) and yields optimization trajectories with explicit topological interpretability.
📝 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.