Author

Yuankai Teng

Date of Award

Summer 2023

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Lili Ju

Second Advisor

Zhu Wang

Abstract

Deep learning has emerged as a powerful approach for solving complex problems in scientific computing due to the increasing availability of large-scale data and computational resources. This thesis explores the potential of deep learning methods for three specific problems in scientific computing: (i) reducing the dimensions of variables in function approximation, (ii) solving linear reaction-diffusion equations, and (iii) finding the parametric representations of parameters in the numerical schemes for solving time-dependent partial differential equations.

For the first problem, a novel deep learning architecture is developed for reducing the dimensions of variables in function approximation. The proposed method achieves state-of-the-art performance on given testing examples and real-world related problems. For the second problem, a deep learning-based method using a fully connected network structure is proposed for solving linear reaction-diffusion equations. The proposed method outperforms existing numerical methods on benchmark problems. For the third problem, a deep learning framework using convolutional neural networks is developed for finding parametric representations of parameters in numerical schemes for solving time-dependent PDEs. The proposed method is shown to be effective on various problems.

The contributions of this thesis have significant implications for advancing the state-of-the-art in scientific computing, enabling more accurate and efficient solutions to complex problems in science and engineering. The proposed deep learning methods provide a promising alternative to existing numerical methods for solving the considered problems and have the potential to greatly accelerate scientific discovery and innovation.

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