一种改进的基于深度神经网络的偏微分方程求解方法

偏微分方程求解是计算流体力学等科学与工程领域中数值分析的计算核心。由于物理的多尺度特性和对离散网格质量的敏感性，传统的数值求解方法通常包含复杂的人机交互和昂贵的网格剖分开销，限制了其在许多实时模拟和优化设计问题上的应用效率。提出了一种改进的基于深度神经网络的偏微分方程求解方法TaylorPINN。该方法利用深度神经网络的万能逼近定理和泰勒公式的函数拟合能力，实现了无网格的数值求解过程。在Helmholtz、Klein-Gordon和Navier-Stokes方程上的数值实验结果表明，TaylorPINN能够很好地拟合计算域内时空点坐标与待求函数值之间的映射关系，并提供了准确的数值预测结果。与常用的基于物理信息神经网络方法相比，对于不同的数值问题,TaylorPINN将预测精度提升了3~20倍。

An improved method for solving partialdifferential equations using deep neural networks

Solving partial differential equations plays a vital role of numerical analysis in scientific and engineering fields such as computational fluid dynamics. Due to the multi-scale nature of physics and sensitivity to the quality of the discrete mesh, traditional numerical methods often require complex human-computer interaction and expensive meshing overhead, which limit their application to many real-time simulation and optimal design problems. This paper proposes an improved neural network-based method for solving partial differential equations, named TaylorPINN. It utilizes the universal approximation theorem of neural networks and the function-fitting capability of Taylor formula, and provides a mesh-free numerical solving process. Numerical experimental results on Helmholtz, Klein-Gordon, and Navier-Stokes equations demonstrate that TaylorPINN is able to approximate the underlying mapping relations between the coordinate inputs and quantities of interest, yielding an accurate prediction result. Compared with the widely used physics-informed neural network method, TaylorPINN improves the prediction accuracy by a factor of 3~20x across different numerical problems.