확률미분방정식의 해를 구하기 위한 다항카오스 구조의 적대적 생성 신경망

Abstract

In this doctoral thesis, I present deep-learning-based approaches for solving stochastic partial differential equations (SPDEs) and explore the use of polynomial chaos structure in numerical analysisapplied to the deep learning model for reduced time and memory costs. To this end, I first introduce the Karhunen-Loève expansion, which approximates a stochastic process by using a set of random variables, and discuss the generalized polynomial chaos method for solving the approximated SPDEs, addressing thecurse of dimensionality that frequently emerges in high-dimensional problems. Next, I introduce physics-informed neural network (PINN), a deep learning model designed for solving differential equations, and elaborate on the physics-informed generative adversarial networks (GANs), which combines the methodologies employed in PINNs and GANs to effectively solve SPDEs. In particular, I highlight that incorporating the polynomial chaos structure into the generative model can significantly reduce the computational costs without compromising the accuracy of the existing models. Furthermore, numerical examples will be demonstrated that when solving an SPDE involving multiple independent stochastic processes, applying distinct discriminators separately can enhance the accuracy of the model.