D. L. Young

Takemi Shigeta, D. L. Young

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.

Amir Noorizadegan, D. L. Young, Y. C. Hon et al.

In this study, we investigate how the updating of weights during forward operation and the computation of gradients during backpropagation impact the optimization process, training procedure, and overall performance of the neural network, particularly the multi-layer perceptrons (MLPs). This paper introduces a novel neural network structure called the Power-Enhancing residual network, inspired by highway network and residual network, designed to improve the network's capabilities for both smooth and non-smooth functions approximation in 2D and 3D settings. By incorporating power terms into residual elements, the architecture enhances the stability of weight updating, thereby facilitating better convergence and accuracy. The study explores network depth, width, and optimization methods, showing the architecture's adaptability and performance advantages. Consistently, the results emphasize the exceptional accuracy of the proposed Power-Enhancing residual network, particularly for non-smooth functions. Real-world examples also confirm its superiority over plain neural network in terms of accuracy, convergence, and efficiency. Moreover, the proposed architecture is also applied to solving the inverse Burgers' equation, demonstrating superior performance. In conclusion, the Power-Enhancing residual network offers a versatile solution that significantly enhances neural network capabilities by emphasizing the importance of stable weight updates for effective training in deep neural networks. The codes implemented are available at: \url{https://github.com/CMMAi/ResNet_for_PINN}.

Takemi Shigeta, D. L. Young

The method of fundamental solution (MFS) is an efficient meshless method for solving a boundary value problem in an exterior unbounded domain. The numerical solution obtained by the MFS is accurate, while the corresponding matrix equation is ill-conditioned. A modified MFS (MMFS) with the proper basis functions is proposed by the introduction of the modified Trefftz method (MTM). The concrete expressions of the corresponding condition numbers and the solvability by these methods are mathematically proven. Thereby, the optimal parameter minimizing the condition number is also mathematically given. Numerical experiments show that the condition numbers of the matrices corresponding to the MTM and the MMFS are reduced and that the numerical solution by the MMFS is more accurate than the one by the conventional method.

A. Noorizadegan, R. Cavoretto, D. L. Young et al.

Background: Deep learning techniques, particularly neural networks, have revolutionized computational physics, offering powerful tools for solving complex partial differential equations (PDEs). However, ensuring stability and efficiency remains a challenge, especially in scenarios involving nonlinear and time-dependent equations. Methodology: This paper introduces novel residual-based architectures, namely the Simple Highway Network and the Squared Residual Network, designed to enhance stability and accuracy in physics-informed neural networks (PINNs). These architectures augment traditional neural networks by incorporating residual connections, which facilitate smoother weight updates and improve backpropagation efficiency. Results: Through extensive numerical experiments across various examples including linear and nonlinear, time-dependent and independent PDEs we demonstrate the efficacy of the proposed architectures. The Squared Residual Network, in particular, exhibits robust performance, achieving enhanced stability and accuracy compared to conventional neural networks. These findings underscore the potential of residual-based architectures in advancing deep learning for PDEs and computational physics applications.

A. Noorizadegan, Y. C. Hon, D. L. Young et al.

Surface reconstruction from point clouds is a fundamental challenge in computer graphics and medical imaging. In this paper, we explore the application of advanced neural network architectures for the accurate and efficient reconstruction of surfaces from data points. We introduce a novel variant of the Highway network (Hw) called Square-Highway (SqrHw) within the context of multilayer perceptrons and investigate its performance alongside plain neural networks and a simplified Hw in various numerical examples. These examples include the reconstruction of simple and complex surfaces, such as spheres, human hands, and intricate models like the Stanford Bunny. We analyze the impact of factors such as the number of hidden layers, interior and exterior points, and data distribution on surface reconstruction quality. Our results show that the proposed SqrHw architecture outperforms other neural network configurations, achieving faster convergence and higher-quality surface reconstructions. Additionally, we demonstrate the SqrHw's ability to predict surfaces over missing data, a valuable feature for challenging applications like medical imaging. Furthermore, our study delves into further details, demonstrating that the proposed method based on highway networks yields more stable weight norms and backpropagation gradients compared to the Plain Network architecture. This research not only advances the field of computer graphics but also holds utility for other purposes such as function interpolation and physics-informed neural networks, which integrate multilayer perceptrons into their algorithms.