  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.
