HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations
这是一篇使用同调法与 PINN 相结合解决非线性椭圆微分方程的论文，并处理了不规则边界区域。
摘要
摘要原文：
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.
摘要翻译：
基于物理信息神经网络（PINNs）的机器学习是一种新兴的非线性微分方程求解框架。然而，由于神经网络结构的隐含规律性，PINNs 在大多数情况下只能通过最小化损失函数找到最平坦的解。在本文中，我们将 PINNs 与同调延续法（一种计算多项式系统孤立根的经典数值方法）相结合，提出了一种新的深度学习框架，命名为同调物理信息神经网络（HomPINNs），用于求解非线性椭圆微分方程的多解。HomPINN 的实现是一个同调过程，由一个名为起始神经网络的全连接神经网络的训练和多个具有不同跟踪参数的 PINN 的训练过程组成。起始神经网络用于逼近由三元解构建的起始函数，而其他 PINN 则用于最小化由边界条件和同调函数定义的损失函数，这些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调过程的不同步骤，一个 PINN 由上一步训练有素的神经网络初始化，而第一个起始神经网络则使用默认初始化方法初始化。本文列举了几个数值示例来说明我们提出的 HomPINN 的效率，其中包括具有心形域的反应扩散方程。