物理信息神经网络求解纳维–斯托克斯方程组正反问题

Physics-Informed Neural Networks for Solving Forward and Inverse Problems in Navier-Stokes Equations

摘要: 基本文提出了一种基于物理信息神经网络的求解器，用于求解纳维–斯托克斯方程组的正反问题。该方法结合深度神经网络与自动微分技术，直接以空间和时间坐标作为输入，预测流场的速度和压力变量。此求解器的核心在于构建一个包含物理方程残差(动量方程与连续性方程)、边界条件损失和初始条件损失的多目标损失函数，并通过引入权重系数平衡各项贡献。网络采用双曲正切函数作为激活函数以保证高阶导数的计算需求，并利用Adam和L-BFGS-B优化器进行训练。该求解器具有无需网格生成的优势。数值实验证明了求解器的有效性和鲁棒性。

Abstract: This paper proposes a Physics-Informed Neural Network (PINN)-based solver for solving forward and inverse problems in the Navier-Stokes equations. The method integrates deep neural networks with automatic differentiation techniques, directly taking spatial and temporal coordinates as inputs to predict fluid flow velocity and pressure fields. The core methodology involves constructing a multi-objective loss function that incorporates residuals of the governing physics equations (momentum and continuity equations), boundary condition loss, and initial condition loss. Weight coefficients are introduced to balance the contributions of these components. The network utilizes the hyperbolic tangent (tanh) activation function to accommodate the computational requirements for high-order derivatives and is trained using the Adam and L-BFGS-B optimizers. This solver eliminates the need for mesh generation. Numerical experiments validate the effectiveness and robustness of the proposed solver.