论文标题
多尺度问题均质化的神经网络方法
A Neural Network Approach for Homogenization of Multiscale Problems
论文作者
论文摘要
我们提出了一种基于神经网络的方法来实现多尺度问题的均质化。所提出的方法使用训练损失的无衍生化公式，该公式结合了布朗步行者，以找到多尺度PDE溶液的宏观描述。与其他基于网络的问题的方法相比，所提出的方法没有手工制作的神经网络架构的设计和细胞问题，以计算均化系数。布朗步行者的探索社区会影响整体学习轨迹。我们通过神经网络确定了分别捕获局部异质和全局均匀溶液行为的微观和宏观步骤的界限。边界表明，所提出方法的计算成本独立于标准周期性问题的微观周期性结构。我们通过周期性和随机场系数的线性和非线性多尺度问题来验证所提出的方法的效率和鲁棒性。
We propose a neural network-based approach to the homogenization of multiscale problems. The proposed method uses a derivative-free formulation of a training loss, which incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution. Compared with other network-based approaches for multiscale problems, the proposed method is free from the design of hand-crafted neural network architecture and the cell problem to calculate the homogenization coefficient. The exploration neighborhood of the Brownian walkers affects the overall learning trajectory. We determine the bounds of micro- and macro-time steps that capture the local heterogeneous and global homogeneous solution behaviors, respectively, through a neural network. The bounds imply that the computational cost of the proposed method is independent of the microscale periodic structure for the standard periodic problems. We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems with periodic and random field coefficients.
