Emulation or surrogate modeling is an indispensable part of many scientific and engineering disciplines. However, most emulators such as Gaussian processes (GPs) or deep neural networks (DNNs) are exclusively developed for interpolation/regression and have limited generalization power. In this thesis we present two approaches for improved and scalable emulation of physical systems.
In the context of finding governing dynamics for a physical system, we introduce evolutionary Gaussian processes (EGPs). EGPs are a systematic integration of GPs and evolutionary programming (EP) that boost the performance of GPs by the automatic discovery of free-form symbolic bases that regress the data. As we demonstrate via examples that include a host of analytical functions as well as an engineering problem on materials modeling, EGP can improve the performance of ordinary GPs in terms of not only extrapolation, but also interpolation/regression and numerical stability.
In the context of surrogating solvers for PDEs, we introduce a transferable framework for solving intial conditions and boundary value problems (IBVPs) via DNNs which can be trained once and used forever for various domains of unseen sizes, shapes, and boundary conditions. We present the genomic flow network (GFNet), a neural network that can infer the solution of an IBVP subject to arbitrary initial and boundary conditions on a square domain called genome. Furthermore, we propose mosaic flow (MF) predictor, a novel iterative algorithm that assembles the GFNet's inferences for IBVPs on large domains with unseen sizes and shapes while preserving the spatial regularity of the solution.