2 posts tagged with "Stokes 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。

这是一篇变系数的波动方程，提出了一种新的格式。

摘要​

摘要原文：

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

摘要翻译：

一种新型高效的神经网络与有限差分混合方法被开发用于求解具有嵌入式不规则界面跳变不连续性的规则域中的泊松方程。由于解在界面处具有较低的正则性，当对该问题应用有限差分离散化时，必须采用额外处理以考虑跳变不连续性。本文旨在通过机器学习方法将此额外处理简化，以提升实现效率。

核心思想是将解分解为奇异部分和规则部分。神经网络学习机制结合给定的跳变条件求解奇异解，而标准五点拉普拉斯离散化用于获得满足边界条件的规则解。无论界面几何如何，这两个任务仅需监督学习进行函数逼近和快速直接求解器求解泊松方程，使混合方法易于实现且高效。二维和三维数值结果表明，本混合方法可保持解及其导数的二阶精度，且与文献中传统的浸入式界面方法相当。作为应用示例，我们通过求解带有奇异力的斯托克斯方程，验证了本方法的鲁棒性。