SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2581-A2613, January 2021.
We propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying Kolmogorov $n$-width. However, these networks are restricted to data on structured meshes. Unstructured meshes are often required for performing analyses of real systems with complex geometry. Our custom graph convolution operators based on the available differential operators for a given spatial discretization effectively extend the application space of deep convolutional autoencoders to systems with arbitrarily complex geometry that are typically discretized using unstructured meshes. We propose sets of convolution operators based on the spatial derivative operators for the underlying spatial discretization, making the method particularly well suited to data arising from the solution of partial differential equations. We demonstrate the method using examples from heat transfer and fluid mechanics and show better than an order of magnitude improvement in accuracy over linear methods.

中文翻译：

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使用非结构化空间离散化计算物理数据非线性流形学习的定制卷积神经网络

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2581-A2613 页，2021 年 1 月。
我们提出了一种基于深度卷积自动编码器的非线性流形学习技术，适用于复杂几何中物理系统的模型降阶。卷积神经网络已被证明对于压缩由系统产生的数据非常有利，这些系统展示了缓慢衰减的 Kolmogorov $n$-width。但是，这些网络仅限于结构化网格上的数据。对具有复杂几何形状的真实系统进行分析通常需要非结构化网格。我们基于给定空间离散化的可用微分算子的自定义图卷积算子有效地将深度卷积自动编码器的应用空间扩展到具有任意复杂几何形状的系统，这些几何形状通常使用非结构化网格进行离散化。我们为底层空间离散化提出了基于空间导数算子的卷积算子集，使该方法特别适合于偏微分方程解产生的数据。我们使用来自传热和流体力学的示例演示了该方法，并显示出比线性方法的精度提高了一个数量级。