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.

翻译：摘要：我们提出了一种用于深度学习架构的新型卷积层，称为QuadConv——通过求积法逼近连续卷积的算子。该算子专门针对基于网格的非均匀数据设计，通过学习可在任意位置采样的连续核函数实现这一目标。此外，我们详细阐述并构建了该算子的高效实现方法。为验证算子性能，我们考虑对固定网格上的偏微分方程（PDE）仿真数据进行压缩任务。通过将QuadConv自编码器（QCAE）与标准卷积自编码器（CAE）进行对比，我们证明QuadConv在均匀网格数据上能够达到与标准离散卷积相当的性能。进一步研究表明，即使在非均匀数据上，QCAE仍能保持同等精度。在这两种情况下，QuadConv均优于图卷积等其他非结构化卷积方法。