Bi-fidelity informed Gaussian process regression methods and their applications

Parameterized partial differential equations are widely applied in engineering and the applied sciences, such as in acoustic systems, solid and fluid mechanics, and financial problems. Parameterized partial differential equations are typically solved using traditional numerical methods, which can be computationally expensive. Alternatively, the reduced basis methods have become popular methods for this task due to their efficiency. Yet, the traditional reduced basis method may require rewrites of the original solver, which is a challenging task. In addition, the traditional reduced basis methods can be inefficient for solving nonlinear problems. The goal of this thesis is to construct a noninstrusive method for approximating the solutions of nonlinear parameterized PDEs with a limited amount of available high-fidelity data. To this end, the Gaussian Process (Kriging) is a preferred approach, particularly when a small amount of data is available. However, most existing literature only uses single-fidelity data. Specifically, high-fidelity data has higher accuracy at a high computational cost. In contrast, low-fidelity data is computationally cheaper but less accurate. This thesis intends to use Bi-fidelity data, which can reach a comparable level of accuracy as high-fidelity data at a lower computational cost. The outline of this thesis is first to construct a Bi-fidelity informed Kriging to approximate the solutions of the nonlinear parameterized PDEs. Contrary to [52], we construct Bi-fidelity input features by combining parameters from high-fidelity data and low-fidelity data coefficients. The second part of the thesis aims to construct Bi-fidelity Weighted Transfer Learning (BFWL) [18] for approximating function values of high-fidelity data. The traditional machine learning for each task is independent, even though they are related. While transfer learning enhances the learning process in new tasks by leveraging the knowledge from the related tasks they already learned. In addition, the Cokriging method extends the traditional Kriging method to multi-fidelity problems and leverages the covariance between two correlated variables. Given the advantage of Cokriging [96] and concepts of transfer learning [18], this thesis applies Bi-fidelity weighted transfer learning (BFWL) [18] with Cokriging method.