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FD-PINN: 频域物理信息神经网络

FD-PINN: FREQUENCY DOMAIN PHYSICS-INFORMED NEURAL NETWORK

  • 摘要: 物理信息神经网络(physics-informed neural network, PINN)是将模型方程编码到神经网络中, 使网络在逼近定解条件或观测数据的同时最小化方程残差, 实现偏微分方程求解. 该方法虽然具有无需网格划分、易于融合观测数据等优势, 但目前仍存在训练成本高、求解精度低等局限性. 文章提出频域物理信息神经网络(frequency domain physics-informed neural network, FD-PINN), 通过从周期性空间维度对偏微分方程进行离散傅里叶变换, 偏微分方程被退化为用于约束FD-PINN的频域中维度更低的微分方程组, 该方程组内各方程不仅具有更少的自变量, 并且求解难度更低. 因此, 与使用原始偏微分方程作为约束的经典PINN相比, FD-PINN实现了输入样本数目和优化难度的降低, 能够在降低训练成本的同时提升求解精度. 热传导方程、速度势方程和Burgers方程的求解结果表明, FD-PINN普遍将求解误差降低1 ~ 2个数量级, 同时也将训练效率提升6 ~ 20倍.

     

    Abstract: 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.

     

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