We present a structure preserving PINN for solving a series of time dependent PDEs with periodic boundary. Our method can incorporate the periodic boundary condition as the natural output of any deep neural net, hence significantly improving the training accuracy of baseline PINN. Together with mini-batching and other PINN variants (SA-PINN, RBA-PINN, etc.), our structure preserving PINN can even handle stiff PDEs for modeling a wide range of convection-diffusion and reaction-diffusion processes. We demonstrate the effectiveness of our PINNs on various PDEs from Allen Cahn, Gray Scott to nonlinear Schrodinger.

翻译：本文提出了一种保持结构守恒的物理信息神经网络（PINN），用于求解一系列带周期边界条件的时变偏微分方程。该方法可将周期边界条件作为任意深度神经网络的天然输出进行嵌入，从而显著提升基线PINN的训练精度。结合小批量训练及其他PINN变体（如SA-PINN、RBA-PINN等），我们的结构守恒PINN甚至能够处理刚性偏微分方程，用于模拟广泛的对流扩散及反应扩散过程。我们在Allen-Cahn方程、Gray-Scott方程以及非线性薛定谔方程等多个偏微分方程上验证了所提PINN方法的有效性。