论文标题
具有不确定参数的PDE的多级神经网络
Multi-level neural networks for PDEs with uncertain parameters
论文作者
论文摘要
提出了一种具有不确定参数的偏微分方程的新型多级方法。该方法的原理是，多级方法中的网格级别之间的误差具有空间结构，该空间结构是良好的近似值，而与实际的网格级别无关。我们的方法通过采用一系列卷积神经网络来学习这种结构，这些卷积神经网络非常适合自动检测局部误差特征作为解决方案的潜在量。此外，通过使用转移学习的概念，将粗网格水平的信息重复使用，以最大程度地减少精细水平的样品数量。该方法的表现优于最先进的多级方法，尤其是在涉及复杂的PDE（例如单相和自由表面流问题）的情况下，或者需要高精度时。
A novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good approximation independent of the actual grid level. Our method learns this structure by employing a sequence of convolutional neural networks, that are well-suited to automatically detect local error features as latent quantities of the solution. Furthermore, by using the concept of transfer learning, the information of coarse grid levels is reused on fine grid levels in order to minimize the required number of samples on fine levels. The method outperforms state-of-the-art multi-level methods, especially in the case when complex PDEs (such as single-phase and free-surface flow problems) are concerned, or when high accuracy is required.
