Deep data-driven modeling and control of high-dimensional nonlinear systems
Keywords: Aeronautics and Astronautics.

The ability to derive models for dynamical systems is a central focus in many realms of science and engineering. However, for many systems of interest, the governing equations are either unknown or can only be evaluated to high accuracy at significant computational expense. Difficulties with modeling can be further exacerbated by additional complexities, such as high-dimensional states or nonlinearities in the dynamics. In turn, these challenges can hinder performance on important downstream tasks, such as prediction and control. This thesis presents techniques for learning dynamics models from data. By taking a data-driven approach, models can be derived even for systems with governing equations that are unknown or expensive to evaluate. Furthermore, training procedures can be tailored to provide learned models with desirable properties, such as low dimensionality (for efficient evaluation and storage) or linearity (for control). The proposed techniques are primarily evaluated on their ability to learn from data generated by computational fluid dynamics (CFD) simulations. CFD data serves as an ideal test case for data-driven techniques because the simulated fluid flows are nonlinear and can exhibit a wide array of behaviors. Additionally, modeling and even storage of CFD data can prove challenging due to the large number of degrees of freedom in many simulations, which can cause time snapshots of the flow field to contain megabytes or even gigabytes of data. First, this thesis proposes a multi-stage compression procedure to alleviate the storage overhead associated with running large-scale CFD simulations. Individual time snapshots are compressed through a combination of neural network autoencoders and principal component analysis. Subsequently, a dynamics model is learned that can faithfully propagate the compressed representations in time. The proposed method is able to compress the stored data by a factor of over a million, while still allowing for accurate reconstruction of all flow solutions at all time instances. The high computational cost of CFD simulations can make it impractical to run large numbers of simulations at diverse flow conditions. The second part of this thesis introduces a method for performing generative modeling, which allows for the efficient simulation of fluid flows at a wide range of flow conditions given data from only a subset of those conditions. The proposed method, which relies upon techniques from variational inference, is shown to generate accurate simulations at a range of conditions for both two- and three-dimensional fluid flow problems. The equations that govern fluid flow are nonlinear, meaning that many control techniques, largely derived for linear systems, prove ineffective when applied to fluid flow control. This thesis proposes a method, grounded in Koopman theory, for discovering data-driven linear models that can approximate the forced dynamics of systems with nonlinear dynamics. The method is shown to produce stable dynamics models that can accurately predict the time evolution of airflow over a cylinder. Furthermore, by performing model predictive control with the learned models, a straightforward, interpretable control law is found that is capable of suppressing vortex shedding in the cylinder wake. In the final part of this thesis, the Deep Variational Koopman (DVK) model is introduced, which is a method for inferring distributions over Koopman observations that can be propagated linearly in time. By sampling from the inferred distributions, an ensemble of dynamics models is obtained, which in turn provides a distribution over possible outcomes as a modeled system advances in time. Experiments show that the DVK model is capable of accurate, long-term prediction for a variety of dynamical systems. Furthermore, it is demonstrated that accounting for the uncertainty present in the distribution over dynamics models enables more effective control.