Analysis of computer experiments with applications in material science, robotics and inverse problem
Creators: name: Song, Yao
Publisher: name: No Publisher Supplied
Year: 2023

description: Computer experiments have emerged as a powerful approach for studying real-world systems through the utilization of complex mathematical models. These experiments have gained considerable traction as viable alternatives to physical experiments, particularly in the investigation of intricate systems in science and engineering domains. Unlike physical experiments, computer experiments can be conducted at a fraction of the cost and offer the advantage of being nearly deterministic. That is, when provided with the same input, the computer experiment will yield highly consistent outputs across multiple runs. In light of these characteristics, the development of efficient statistical emulators for computer experiments becomes crucial. An emulator serves as an interpolator, capturing the relationship between the inputs and outputs of the computer experiment. By constructing an accurate emulator, we can effectively mimic the behavior of the actual computer experiment and use it to make predictions for untested input configurations. Additionally, emulators offer the valuable capability of providing prediction uncertainty, aiding decision-making processes in real-world applications. One statistical technique that has gained significant prominence in this domain is Gaussian process modeling. Gaussian processes provide a flexible and powerful framework for capturing the underlying relationships between input variables and corresponding system outputs. By leveraging the inherent properties of Gaussian processes, researchers can effectively emulate computer experiments, estimate prediction uncertainties, and explore the response surface of complex systems. This thesis aims to address the challenge of building robust and efficient statistical emulators for computer experiments, accompanied by the estimation and quantification of prediction uncertainty. By leveraging advanced statistical methodologies, such as Gaussian Process and Machine Learning Methods, we seek to develop techniques that capture the complex relationships between inputs and outputs. This comprehensive understanding of complex systems will facilitate profound insights into their behavior, thereby empowering us to drive enhanced efficiency in diverse domains such as design, optimization, and decision-making processes. The first paper in Chapter 1 focuses on the optimized exploration problem of force-field systems. It introduces an approach that combines machine learning techniques with smart experimental design ideas to identify multiple favorable regions in the input space. This structured framework comprises five essential steps and demonstrates its efficacy through a comprehensive simulation analysis on two ReaxFF systems to explore a deterministic multivariate response surface in material science and identify points in the ``desired" regions. The second paper in Chapter 2 presents an integration of surrogate modeling and topology for reducing the amount of data required to describe the global dynamics of a closed-box robot controller. By employing randomized short trajectories and a Gaussian Process surrogate model, the paper constructs a combinatorial representation known as the "Morse graph." This representation captures the control system's attractors, regions of attraction, and provides a pointwise confidence level for global dynamics estimation. Notably, this framework overcomes the need for high prediction accuracy and outperforms existing analytical and machine learning alternatives in terms of data efficiency without sacrificing accuracy. The third paper in Chapter 3 addresses the challenging task of functional inverse scattering, where the goal is to infer the refractive index of a bounded scatterer. The paper proposes a novel Bayesian framework that utilizes a surrogate model incorporating functional inputs through kernel functions. By employing a Bayesian procedure, the framework infers functional inputs by leveraging the posterior distribution. Moreover, the Bayesian framework is extended to integrate multi-fidelity simulations, combining high-fidelity finite element methods with the low-fidelity Born approximation. This extension enables more accurate functional recoveries and provides quantification of uncertainty compared to existing methods relying on finite basis expansion. Collectively, these three papers contribute to the advancement of statistical modeling, machine learning, and dynamical systems research. By addressing critical challenges in complex multivariate exploration, dynamical system modeling, and functional inverse inference, these works offer valuable insights and innovative methodologies that contribute to the broader field of computer experiment.