HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations
HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations
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DOI：
10.1016/j.camwa.2022.07.002
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发表时间：
2022-07-20
影响因子：
2.9
通讯作者：
Lin, Guang
中科院分区：
文献类型：
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作者：
Huang, Yao;Hao, Wenrui;Lin, Guang
基于物理信息神经网络（PINNs）的机器学习是解决非线性微分方程的新兴框架。然而，由于神经网络结构的隐含规律性，PINNs只能找到最平坦的解决方案，在大多数情况下，通过最小化损失函数。在本文中，我们将联合收割机PINNs与同伦连续方法（一种计算多项式系统孤立根的经典数值方法）相结合，并提出了一种新的深度学习框架，称为同伦物理信息神经网络（HomPINNs），用于求解非线性椭圆微分方程的多解。HomPINN的实现是一个同伦过程，它由一个完全连接的神经网络（称为起始神经网络）的训练和具有不同跟踪参数的几个PINN的训练过程组成。起始神经网络用于逼近由平凡解构造的起始函数，而其他PINN用于最小化由边界条件和同伦函数定义的损失函数，随不同的跟踪参数而变化。这些训练过程被重新划分为同伦过程的不同步骤，PINN由前一步训练好的神经网络初始化，而第一个起始神经网络使用默认初始化方法初始化。几个数值例子来显示我们提出的HomPINNs的效率，包括反应扩散方程的心形区域。
Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain.