论文标题
Helmholtz方程式的多移民增强的深度学习预处理
Multigrid-augmented deep learning preconditioners for the Helmholtz equation
论文作者
论文摘要
在本文中，我们提出了一种数据驱动的方法，用于迭代地求解高波数的离散异质Helmholtz方程。在我们的方法中，我们将经典迭代求解器与卷积神经网络（CNN）相结合，形成了在Krylov求解器中应用的预处理。对于预处理，我们使用与多族成分结合使用的U-NET类型的CNN。提出了两种类型的预处理1）U-NET作为粗网格求解器，而2）U-net作为带有laplacian V-Cycles偏移的放气操作员。遵循我们的培训方案和数据实践，我们的CNN预处理可以概括为残差和相对一般的波浪慢模型集。最重要的是，我们还提供了一个编码器固定框架，其中“编码器”网络在介质上概括，并将上下文向量发送到另一个“求解器”网络，该网络概括了右侧。我们表明，此选项比独立变体更强大，更高效。最后，我们还提供了一个迷你擦拭程序，以在已知模型后改善求解器。在求解多个右侧（例如在反问题中）时，此选项是有益的。我们证明了我们方法在各种2D问题上的效率和概括能力。
In this paper, we present a data-driven approach to iteratively solve the discrete heterogeneous Helmholtz equation at high wavenumbers. In our approach, we combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver. For the preconditioner, we use a CNN of type U-Net that operates in conjunction with multigrid ingredients. Two types of preconditioners are proposed 1) U-Net as a coarse grid solver, and 2) U-Net as a deflation operator with shifted Laplacian V-cycles. Following our training scheme and data-augmentation, our CNN preconditioner can generalize over residuals and a relatively general set of wave slowness models. On top of that, we also offer an encoder-solver framework where an "encoder" network generalizes over the medium and sends context vectors to another "solver" network, which generalizes over the right-hand-sides. We show that this option is more robust and efficient than the stand-alone variant. Lastly, we also offer a mini-retraining procedure, to improve the solver after the model is known. This option is beneficial when solving multiple right-hand-sides, like in inverse problems. We demonstrate the efficiency and generalization abilities of our approach on a variety of 2D problems.
