Sun Kuo, Feng Xinlong
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China.
Entropy (Basel). 2023 Apr 18;25(4):674. doi: 10.3390/e25040674.
Physics-informed neural networks (PINNs) are effective for solving partial differential equations (PDEs). This method of embedding partial differential equations and their initial boundary conditions into the loss functions of neural networks has successfully solved forward and inverse PDE problems. In this study, we considered a parametric light wave equation, discretized it using the central difference, and, through this difference scheme, constructed a new neural network structure named the second-order neural network structure. Additionally, we used the adaptive activation function strategy and gradient-enhanced strategy to improve the performance of the neural network and used the deep mixed residual method (MIM) to reduce the high computational cost caused by the enhanced gradient. At the end of this paper, we give some numerical examples of nonlinear parabolic partial differential equations to verify the effectiveness of the method.
物理信息神经网络（PINNs）在求解偏微分方程（PDEs）方面很有效。这种将偏微分方程及其初始边界条件嵌入神经网络损失函数的方法成功地解决了正向和反向PDE问题。在本研究中，我们考虑了一个参数化光波方程，使用中心差分对其进行离散化，并通过这种差分格式构建了一种名为二阶神经网络结构的新神经网络结构。此外，我们使用自适应激活函数策略和梯度增强策略来提高神经网络的性能，并使用深度混合残差法（MIM）来降低由增强梯度导致的高计算成本。在本文结尾，我们给出了一些非线性抛物型偏微分方程的数值例子，以验证该方法的有效性。
