Data-Driven Learning and Optimization of Dynamical Systems

Dynamical systems analysis and optimization is pivotal for safe, efficient, and resilient processes that consistently deliver high-quality products. Conventionally, decision-making and systems behavior analysis have relied heavily on physics-based models. However, these physics-based models pose several challenges to systems analysis and decision-making. These include prohibitively high computational costs associated with the numerical computations of the governing equations, limitations of our understanding of the system's underlying physical mechanisms that lead to insufficient and erroneous predictions, as well as the inherent nonlinearity and stochasticity that real-world systems exhibit. This thesis seeks to address these challenges by developing and applying data-driven methodologies for the dynamic analysis and optimization of complex systems, with emphasis on chemical and biochemical systems. The research presented in this dissertation can be distilled into three main contributions: The first contribution revolves around the utilization of data-driven methods for approximating the dynamic behavior of dynamical systems and the applications that such data-driven models enable. In particular, the thesis focuses on data-driven strategies for learning the dynamics of systems under varying inputs, which can then be employed for uncertainty quantification analysis, optimal experiment design, and real-time decision making. This topic, though extensively investigated in the literature, remains challenging due to the limited, sparse, and noisy nature of available data. Our approach is rooted in the concept of flow-maps, which are operators that predict a system's future state based on its current state. We approximate this transition law using a polynomial chaos expansions-based Gaussian Process (GP), a probablistic non-parametric model that allows us to predict the expected behavior of the system while providing uncertainty bounds. Notably, our proposed approach demonstrates exceptional predictive capabilities even in low-data regimes and offers substantial computational savings compared to high-fidelity models for uncertainty quantification of dynamical systems. The second contribution of this thesis is geared towards applying the Bayesian Optimization (BO) framework for the autotuning of general controllers. This challenge is critical in control theory due to the lack of easily derivable closed-form mathematical expressions for the system's closed-loop performance metrics. Limited and noisy closed-loop data can further compound this problem. BO is an ideal candidate method to address this problem, as it leverages the data-efficiency of GPs to create a probabilistic surrogate model to capture the relationship between decision variables and system performance. Through a careful balance of exploration and exploitation, BO strives to identify the globally optimal solution using informative queries from the closed-loop system. In particular, in this work we broaden the BO framework to address two significant aspects related to the autotuning of biochemical systems. Firstly, we tackle the challenge of BO under time-invariant uncertainty by proposing a new method for adversarially robust BO. This method concurrently learns the mapping from decision variables and uncertainties to performance. Secondly, we tackle the ubiquitous problem of multiple conflicting objectives that arises in real-world scenarios. We propose a multi-objective BO scheme in tandem with a data-driven model that encodes any existing information about the system's characteristics and is partially adaptable. The utility and performance improvement induced by such extensions are demonstrated through a bioreactor benchmark case study. The third and final contribution of this thesis is a novel gradient-enhanced BO framework for closed-loop policy search. This advancement is pivotal in optimization problems where both zeroth- and first-order information are available (i.e., both the reward function and its gradient can be observed) during the query process. Traditional BO approaches may overlook the richness of the gradient information, potentially limiting their optimization efficiency. In contrast, our proposed approach, including two alternative methods, accelerates BO convergence by integrating both performance optimization and formal optimality conditions in the proposal of new query points. An important application of this method is in reinforcement learning, as policy-based methods under stochastic policies can provide objective function and gradient observations. In conclusion, this thesis makes significant strides in the domain of data-driven analysis and optimization of dynamical systems, addressing crucial challenges that stem from the scarcity and low quality of data, a common trend in bioprocesses and integrated biomanufacturing systems. Future research can extend the application of these data-driven methods to emerging fields such as deep space bioprocessing. In such novel domains, data-driven models can fundamentally underpin the optimization of end-to-end system design, planning, and control. Furthermore, despite its demonstrated success and the extensions introduced in this work, BO presents significant opportunities for future research. These include managing high-dimensional design spaces and mixed-integer variables, incorporating black-box safety constraints, and leveraging advanced techniques when gradients are available.

Advisor: Mesbah, Ali
Clark, Douglas