论文标题
基于物理信息的神经网络求解进化方程的训练策略
Pre-training strategy for solving evolution equations based on physics-informed neural networks
论文作者
论文摘要
物理知情的神经网络（PINN）是解决时间进化部分微分方程（PDE）的有前途的方法。但是，标准的PINN方法可能无法以强烈的非线性特性或具有高频溶液的PDE求解PDE。物理知情的神经网络（PINN）是解决时间进化部分微分方程（PDE）的有前途的方法。但是，标准的PINN方法可能无法以强烈的非线性特性或具有高频溶液的PDE求解PDE。 PT-PINN方法将整个时间域的困难问题转化为在小子域中定义的相对简单的问题。在小子域中训练的神经网络为大子域或整个时间域的问题提供了神经网络初始化和额外的监督学习数据。通过数值实验，我们证明了PT-PINN成功地求解了具有强非线性和/或高频解决方案的进化PDE，包括强烈的非线性热方程，Allen-CAHN方程，对流方程，具有高频解决方案的对流方程，以及高频解决方案，以及PT-PINN的转化和精度。 PT-PINN方法是解决时间进化PDE的竞争方法。
The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. The PT-PINN method transforms the difficult problem on the entire time domain to relatively simple problems defined on small subdomains. The neural network trained on small subdomains provides the neural network initialization and extra supervised learning data for the problems on larger subdomains or on the entire time-domain. By numerical experiments, we demonstrate that the PT-PINN succeeds in solving the evolution PDEs with strong non-linearity and/or high frequency solutions, including the strongly nonlinear heat equation, the Allen-Cahn equation, the convection equation with high-frequency solutions and so on, and that the convergence and accuracy of the PT-PINN is superior to the standard PINN method. The PT-PINN method is a competitive method for solving the time-evolution PDEs.
