Dynamical System Parameter Path Optimization using Persistent Homology

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.

Hall Effect Thruster Forecasting using a Topological Approach for Data Assimilation

Hall Effect Thrusters (HETs) are electric thrusters that eject heavy ionized gas particles from the spacecraft to generate thrust. Although traditionally they were used for station keeping, recently They have been used for interplanetary space missions due to their high delta-V potential and their operational longevity in contrast to other thrusters, e.g., chemical. However, the operation of HETs involves complex processes such as ionization of gases, strong magnetic fields, and complicated solar panel power supply interactions. Therefore, their operation is extremely difficult to model thus necessitating Data Assimilation (DA) approaches for estimating and predicting their operational states. Because HET's operating environment is often noisy with non-Gaussian sources, this significantly limits applicable DA tools. We describe a topological approach for data assimilation that bypasses these limitations that does not depend on the noise model, and utilize it to forecast spatiotemporal plume field states of HETs. Our approach is a generalization of the Topological Approach for Data Assimilation (TADA) method that allows including different forecast functions. We show how TADA can be combined with the Long Short-Term Memory network for accurate forecasting. We then apply our approach to high-fidelity Hall Effect Thruster (HET) simulation data from the Air Force Research Laboratory (AFRL) rocket propulsion division where we demonstrate the forecast resiliency of TADA on noise contaminated, high-dimensional data.