Date of Award
Open Access Dissertation
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.
Teng, Y.(2023). Deep Learning Methods for Some Problems in Scientific Computing. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/7436