Optimization of Chaotic Dynamical Systems

Many engineered products and processes are simulated using chaotic dynamical systems, Fusion reactors and aircraft are two examples. The capability to optimize these systems would positively impact multiple engineering disciplines and industries. Unfortunately, chaotic dynamical systems are notoriously difficult to optimize, particularly when the performance metric is a time-averaged quantity. Time-averages from chaotic simulations exhibit deterministic "noise," and this noise renders conventional gradient-based optimization algorithms ineffective. The failure of gradient-based methods is especially problematic for shape and topology optimization problems that have many (>100) design variables and parameters. In short, we lack the algorithmic tools to effectively optimize large-scale chaotic systems. To address these needs identified above, this research project will investigate a novel algorithm for the optimization of chaotic dynamical systems: the Linked Ensemble Aggregation Procedure, or LEAP, for short. LEAP, like other ensemble approaches, uses multiple short-time simulations instead of one long simulation in order avoid the infamous "butterfly effect" – where small changes can propagate unpredictably in later stage. However, unlike conventional ensemble methods, the simulations in LEAP are coupled sequentially, which eliminates the "noise" in time-averaged outputs and makes gradient-based optimization possible. The project's research objectives are to i) investigate LEAP's accuracy, parameters, and limitations; ii) seek to improve LEAP's robustness; and iii) generate a solution algorithm suitable for large-scale problems. To increase the broader impact of the research, the project will develop open-source software and educational resources, and will establish a workshop for the optimization of chaotic systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.