QuadConv：基于数值积分的卷积及其在非均匀 PDE 数据压缩中的应用
QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compression
AI总结：
本文提出 QuadConv 卷积层，通过数值积分近似连续卷积以处理非均匀网格数据，并在 PDE 数据压缩任务中验证其性能可匹配标准卷积且优于图卷积。
摘要：
我们提出了一种用于深度学习架构的新型卷积层，称之为 QuadConv——通过数值积分对连续卷积进行近似。该算子专为非均匀、基于网格的数据而设计，通过学习可在任意位置采样的连续核来实现这一目标。此外，该算子的构造支持一种高效实现，我们对其进行了详细说明和构建。作为对该算子的实验验证，我们考虑了压缩来自固定网格的偏微分方程（PDE）仿真数据的任务。我们通过将 QuadConv 自编码器（QCAE）与标准卷积自编码器（CAE）进行比较，表明 QuadConv 在均匀网格数据上可以达到与标准离散卷积相当的性能。进一步地，我们证明 QCAE 即使在非均匀数据上也能保持这种精度。在两种情况下，QuadConv 还优于图卷积等替代的非结构化卷积方法。
英文摘要：
We present a new convolution layer for deep learning architectures which we call QuadConv -- an approximation to continuous convolution via quadrature. Our operator is developed explicitly for use on non-uniform, mesh-based data, and accomplishes this by learning a continuous kernel that can be sampled at arbitrary locations. Moreover, the construction of our operator admits an efficient implementation which we detail and construct. As an experimental validation of our operator, we consider the task of compressing partial differential equation (PDE) simulation data from fixed meshes. We show that QuadConv can match the performance of standard discrete convolutions on uniform grid data by comparing a QuadConv autoencoder (QCAE) to a standard convolutional autoencoder (CAE). Further, we show that the QCAE can maintain this accuracy even on non-uniform data. In both cases, QuadConv also outperforms alternative unstructured convolution methods such as graph convolution.
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