基于梯度增强物理信息神经网络的一维弱噪声随机Burgers方程求解

Solution of One-Dimensional Stochastic Burgers Equation by Gradient-Enhanced Physics-Informed Neural Networks with Weak Noise

摘要: 物理信息神经网络PINNs (Physics-Informed Neural Networks, PINNs)因其良好的可解释性和泛化能力，被当作求解偏微分方程的一类通用逼近器。传统PINNs通过在损失函数中引入方程残差，将方程所携带的物理先验信息编译至网络中，但在求解实际问题时往往需要数量足够大的训练数据集，其输出才具有较好的网络表现力和求解精度。针对一维弱噪声随机Burgers方程，提出梯度增强的物理信息神经网络(Gradient-enhanced Physics-Informed Neural Networks, G-PINNs)，采用方程导数的残差作为一种优化的软约束指导网络迭代更新，理论分析PINNs与G-PINNs两种网络架构寻找方程计算域内最优解的能力。改变数据集的噪声比例和配置点数量，评估两种网络架构拟合一维Burgers方程解结构的性能差异。数值算例表明，G-PINNs在无噪声、1%、3%和5%弱噪声下求解一维Burgers方程的精度分别提高了约80%、44%、31%、和20%。相较于传统PINNs，当迭代次数相同时G-PINNs在无噪声与弱噪声的情况下均能在更少的数据集上拟合一维Burgers方程的解结构，获得精度更高的预测解。

Abstract: Physics-Informed Neural Networks (PINNs) are regarded as a general approximator for solving partial differential equations because of their good interpretability and generalization ability. Traditional PINNs compile the physical prior information carried by the equation into the network by introducing the residual of the equation into the loss function. However, when solving practical problems, it often requires a large number of training datasets to achieve better network performance and solution accuracy. For one-dimensional weak noise random Burgers equation, Gradient-enhanced Physics-Informed Neural Networks (G-PINNs) are proposed. The residual of the derivative of the equation is used as an optimization soft constraint to guide the iterative updating of the network. The ability of PINNs and G-PINNs network architectures to find the optimal solution in the computational domain of the equation is analyzed theoretically. The noise ratio and the number of configuration points of the dataset were changed to evaluate the performance difference of the two network architectures in fitting the solution structure of the one-dimensional Burgers equation. Numerical examples show that the accuracy of G-PINNs in solving one-dimensional Burgers equation with no noise, 1%, 3% and 5% weak noise is improved by about 80%, 44%, 31% and 20%, respectively. Compared with traditional PINNs, G-PINNs can fit the solution structure of one-dimensional Burgers equation on fewer datasets with no noise and weak noise, and obtain a more accurate predictive solution with the same number of iterations.