FD-PINN: 频域物理信息神经网络

Physics-informed neural network (PINN) is a method for solving partial differential equations by encoding model equations into neural network, which fits solutions by simultaneously minimizing equation residuals and approximating definite solution conditions or observation data. Despite the fact that this approach has the benefits of being mesh-free and allowing easy integration of observation data, it still suffers from drawbacks such as high cost of training and limited accuracy in finding solutions. To break these limitation, Frequency domain physics-informed neural network (FD-PINN) is proposed in this paper. The approach involves using discrete Fourier transform on a partial differential equation in the periodic spatial dimension. This transforms the equation into a lower-dimensional system of differential equations in the frequency domain, which are then used to constrain FD-PINN. Due to the fact that each equation within the system of differential equations not only has fewer independent variables, but also has a lower difficulty in solving it. Therefore, compared to the classical PINN using the original partial differential equation as a constraint, the advantage of FD-PINN is that it reduces the number of input samples and the difficulty of optimization, and can improve the solution accuracy while reducing training costs. To demonstrate the effectiveness of FD-PINN, we test it on three different partial differential equations: the heat equation, the Laplace's equation for flow around a cylinder, and the Burgers equation. The results show that FD-PINN generally reduces the solution error by 1-2 orders of magnitude and improves the training efficiency by more than 6 times.