  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}.
