一种求解偏微分方程的动态平衡物理信息神经网络

A dynamic balanced physics-informed neural network for solving partial differential equations

近年来,物理信息神经网络(physics-informed neural networks, PINNs)在求解非线性偏微分方程(partial differential equations, PDEs)中得到了大量应用. PINN将物理信息作为正则化约束加入神经网络损失函数,可以减少传统神经网络方法对训练数据的大量依赖.然而, PINN无法根据数据变化动态调整损失函数中各个损失项的权重,导致其在求解非线性PDEs时存在求解误差较大的问题.为此,本文提出了一种动态平衡物理信息神经网络(dynamic balanced PINN, DBPINN).首先,DBPINN为PINN损失函数的各个损失项设计了一种动态权重系数,并使用随机函数对该系数进行动态更新,能够显著提升PINN的精度.其次, DBPINN为PNNN损失函数的各个损失项之间建立了一种平衡求和方法,该方法考虑了所有损失项之间的竞争关系,使得PINN各损失项朝着有利于收敛的方向进行优化. DBPINN通过动态权重系数和平衡求和方法使得PINN可以更好地进行优化,进而解决了PINN在实际应用中求解误差较大的问题.本文选择了科学机器学习领域中4个经典的非线性PDEs对DBPINN进行了数值验证和分析.实验结果表明,相比于PINN, DBPINN在Schrodinger和Allen-Cahn方程上误差分别降低了46%和64%. DBPINN在求解Navier-Stokes方程时将系数λ_(1)和λ_(2)的误差分别降低了1~2个数量级和约50%. DBPINN在KdV方程中能够在多项系数中将误差降低1个数量级.最后,本文在多种形式的Burgers方程和Allen-Cahn方程上进行性能和参数消融验证,结果表明DBPINN不仅能够提升模型性能、处理小数据量以及拟合不同时间状态下的方程的能力,而且DBPINN相比于PINN具有更好的稳定性、准确率以及收敛性. DBPINN可以取代PINN被应用于各种非线性PDEs的高精度求解.

In recent years,physics-informed neural networks(PINNs)have extensively been used in solving nonlinear partial differential equations(PDEs).PINNs add physical information as a regularization constraint to the neural network loss function,which can reduce the extensive reliance on data by traditional neural network methods.However,this approach makes PINN unable to dynamically adjust the individual residual weights in the loss function according to data changes during training,which leads to the limitation that PINN has large solution errors in solving nonlinear PDEs.In this paper,a Dynamic Balanced PINN(DBPINN)is proposed.Firstly,DBPINN designs a dynamic weight coefficient for each loss term of the PINN loss function,and uses a stochastic function to dynamically update the coeficient,which makes the PINN with only a single loss term converge better.Secondly,DBPINN establishes a balanced summation method for the loss terms of the PNNN loss function,which takes into account the competition among all the loss terms and leads to better convergence of the PINN as a whole.DBPINN makes PINN better optimized by dynamic weighting coefficients and a balanced summation method,which solves the problem of large solution error of PINN in practical applications.In this paper,four classical nonlinear PDEs in the field of scientific machine learning are selected for numerical validation and analysis of DBPINN.The experimental results show that DBPINN reduces the error by 46%and 64%than PINN on the Schrodinger and Allen-Cahn equations,respectively.The DBPINN reduces the error of coefficient λ_(1) by 1 to 2 orders of magnitude and the error of coefficient λ_(2) by about 50%in solving the Navier-Stokes equation,respectively.The DBPINN can reduce the error by 1 order of magnitude in multiple coefficients on the KdV equation.Finally,the performance and parameter ablation are verified on various forms of Burgers and Allen-Cahn equations,and the results show that DBPINN not only improves the model performance,handles small amounts of data,and fits equations in different time states,but also has better stability,accuracy,and convergence than PINN.DBPINN can be used instead of PINN for high accuracy solutions of various nonlinear PDEs.