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Full-Text Articles in Numerical Analysis and Computation
Numerical Approximations Of Phase Field Equations With Physics Informed Neural Networks, Colby Wight
Numerical Approximations Of Phase Field Equations With Physics Informed Neural Networks, Colby Wight
All Graduate Plan B and other Reports, Spring 1920 to Spring 2023
Designing numerical algorithms for solving partial differential equations (PDEs) is one of the major research branches in applied and computational mathematics. Recently there has been some seminal work on solving PDEs using the deep neural networks. In particular, the Physics Informed Neural Network (PINN) has been shown to be effective in solving some classical partial differential equations. However, we find that this method is not sufficient in solving all types of equations and falls short in solving phase-field equations. In this thesis, we propose various techniques that add to the power of these networks. Mainly, we propose to embrace the …
Unitary And Symmetric Structure In Deep Neural Networks, Kehelwala Dewage Gayan Maduranga
Unitary And Symmetric Structure In Deep Neural Networks, Kehelwala Dewage Gayan Maduranga
Theses and Dissertations--Mathematics
Recurrent neural networks (RNNs) have been successfully used on a wide range of sequential data problems. A well-known difficulty in using RNNs is the vanishing or exploding gradient problem. Recently, there have been several different RNN architectures that try to mitigate this issue by maintaining an orthogonal or unitary recurrent weight matrix. One such architecture is the scaled Cayley orthogonal recurrent neural network (scoRNN), which parameterizes the orthogonal recurrent weight matrix through a scaled Cayley transform. This parametrization contains a diagonal scaling matrix consisting of positive or negative one entries that can not be optimized by gradient descent. Thus the …