基于紧支撑径向基函数的物理信息神经网络求解偏微分方程

Compactly Supported Physics-Informed Neural Network for Solving Partial Differential Equations

摘要: 物理信息神经网络(PINN)作为一种融合物理先验知识与深度学习的新型计算范式，在偏微分方程数值求解领域展现出广阔的应用前景。然而，传统PINN采用全连接神经网络作为逼近器，存在谱偏置问题，难以有效捕捉解高频特征和局部变化。针对这一局限性，本文提出一种基于紧支撑径向基函数的单层物理信息神经网络方法(CSPINN)，该方法以紧支撑Wendland函数为基础构建局部化的基函数表示，通过将支撑半径和中心点位置设置为可学习参数，在训练过程中自适应地调整网络的局部感受野，增强网络对解函数高频分量与局部细节的捕捉能力。为增强方法的鲁棒性，本文提出均匀初始化策略，将各中心点的支撑半径初始化为在计算域特征尺度范围内均匀分布的随机值。通过多种类型偏微分方程的数值实验，验证了CSPINN方法的有效性和精度优势。

Abstract: Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations by integrating physical laws into deep learning. However, standard PINNs using multi-layer perceptrons suffer from spectral bias, making it difficult to capture high-frequency features. This paper proposes a Compactly Supported Physics-Informed Neural Network (CSPINN) based on Wendland’s compactly supported radial basis functions. By treating support radii and center locations as learnable parameters, CSPINN adaptively adjusts the local receptive fields during training. A uniform initialization strategy is proposed to enhance robustness. The support radii of each center point are initialized as random values that are uniformly distributed within the characteristic scale range of the computational domain. Numerical experiments on various PDEs demonstrate the effectiveness and accuracy advantages of CSPINN.