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梯度优化物理信息神经网络(GOPINNs):求解复杂非线性问题的深度学习方法

田十方 李彪

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梯度优化物理信息神经网络(GOPINNs):求解复杂非线性问题的深度学习方法

田十方, 李彪

Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving complex nonlinear problems

Tian Shifang, Li Biao
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  • 近年来,物理信息神经网络(PINNs)因其仅通过少量数据就能快速获得高精度的数据驱动解而受到越来越多的关注。然而,尽管该模型在部分非线性问题中有着很好的结果,但它还是有一些不足的地方,例如它的不平衡的反向传播梯度计算导致模型训练期间梯度值剧烈振荡,这容易导致预测精度不稳定。基于此,我们通过梯度统计平衡了模型训练期间损失函数中不同项之间的相互作用,提出了一种梯度优化物理信息神经网络(GOPINNs),该网络结构对梯度波动更具鲁棒性。然后我们以Camassa-Holm(CH)方程、导数非线性薛定谔方程为例,利用GOPINNs模拟了CH方程的peakon解和导数非线性薛定谔方程的有理波解、怪波解。数值结果表明,GOPINNs可以有效地平滑计算过程中损失函数的梯度,并获得了比原始PINNs精度更高的解。总之,我们的工作为优化神经网络的学习性能提供了新的见解,并在求解复杂的CH方程和导数非线性薛定谔方程时用时更少,节约了三分之一多的时间,并且将预测精度提高了将近10倍。
    In recent years, physics-informed neural networks (PINNs) have attracted more and more attention for their ability to quickly obtain high-precision data-driven solutions with only a small amount of data. However, although this model has good results in some nonlinear problems, it still has some shortcomings. For example, the unbalanced back-propagation gradient calculation results in the intense oscillation of the gradient value during the model training, which is easy to lead to the instability of the prediction accuracy. Based on this, we propose a gradient-optimized physics-informed neural networks (GOPINNs) model in this paper, which proposes a new neural network structure and balances the interaction between different terms in the loss function during model training through gradient statistics, so as to make the new proposed network structure more robust to gradient fluctuations. In this paper, taking Camassa-Holm(CH) equation and DNLS equation as examples, GOPINNs is used to simulate the peakon solution of CH equation, the rational wave solution of DNLS equation and the rogue wave solution of DNLS equation. The numerical results show that the GOPINNs can effectively smooth the gradient of the loss function in the calculation process, and obtain a higher precision solution than the original PINNs. In conclusion, our work provides new insights for optimizing the learning performance of neural networks, and saves more than one third of the time in simulating the complex CH equation and the DNLS equation, and improves the prediction accuracy by nearly ten times.
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  • 上网日期:  2023-01-18

梯度优化物理信息神经网络(GOPINNs):求解复杂非线性问题的深度学习方法

  • 宁波大学数学与统计学院, 宁波 315211

摘要: 近年来,物理信息神经网络(PINNs)因其仅通过少量数据就能快速获得高精度的数据驱动解而受到越来越多的关注。然而,尽管该模型在部分非线性问题中有着很好的结果,但它还是有一些不足的地方,例如它的不平衡的反向传播梯度计算导致模型训练期间梯度值剧烈振荡,这容易导致预测精度不稳定。基于此,我们通过梯度统计平衡了模型训练期间损失函数中不同项之间的相互作用,提出了一种梯度优化物理信息神经网络(GOPINNs),该网络结构对梯度波动更具鲁棒性。然后我们以Camassa-Holm(CH)方程、导数非线性薛定谔方程为例,利用GOPINNs模拟了CH方程的peakon解和导数非线性薛定谔方程的有理波解、怪波解。数值结果表明,GOPINNs可以有效地平滑计算过程中损失函数的梯度,并获得了比原始PINNs精度更高的解。总之,我们的工作为优化神经网络的学习性能提供了新的见解,并在求解复杂的CH方程和导数非线性薛定谔方程时用时更少,节约了三分之一多的时间,并且将预测精度提高了将近10倍。

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