This paper proposes a novel neural network framework, denoted as spectral integrated neural networks (SINNs), for resolving three-dimensional forward and inverse dynamic problems. In the SINNs, the spectral integration method is applied to perform temporal discretization, and then a fully connected neural network is adopted to solve resulting partial differential equations (PDEs) in the spatial domain. Specifically, spatial coordinates are employed as inputs in the network architecture, and the output layer is configured with multiple outputs, each dedicated to approximating solutions at different time instances characterized by Gaussian points used in the spectral method. By leveraging the automatic differentiation technique and spectral integration scheme, the SINNs minimize the loss function, constructed based on the governing PDEs and boundary conditions, to obtain solutions for dynamic problems. Additionally, we utilize polynomial basis functions to expand the unknown function, aiming to enhance the performance of SINNs in addressing inverse problems. The conceived framework is tested on six forward and inverse dynamic problems, involving nonlinear PDEs. Numerical results demonstrate the superior performance of SINNs over the popularly used physics-informed neural networks in terms of convergence speed, computational accuracy and efficiency. It is also noteworthy that the SINNs exhibit the capability to deliver accurate and stable solutions for long-time dynamic problems.

翻译：本文提出一种新型神经网络框架——谱集成神经网络（SINNs），用于求解三维正反演动态问题。在SINNs中，采用谱积分方法进行时间离散化，并利用全连接神经网络求解空间域中的偏微分方程（PDEs）。具体而言，网络架构以空间坐标为输入，输出层配置多个输出节点，每个节点专门用于近似谱方法中高斯点所对应不同时间步的解。通过结合自动微分技术与谱积分方案，SINNs最小化由控制偏微分方程和边界条件构建的损失函数，从而获得动态问题的解。此外，我们采用多项式基函数展开未知函数，以提升SINNs处理反问题时的性能。该框架在六个包含非线性偏微分方程的正反演动态问题上进行了测试。数值结果表明，SINNs在收敛速度、计算精度和效率方面均优于广泛使用的物理信息神经网络。值得注意的是，SINNs能够为长时间动态问题提供准确且稳定的解。