QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compression
This provides a domain-specific solution for compressing non-uniform mesh data in PDE simulations, with incremental improvements over existing methods.
The paper tackles the problem of compressing non-uniform PDE simulation data by introducing QuadConv, a quadrature-based convolution layer that learns continuous kernels for arbitrary mesh locations, and shows it matches standard convolutions on uniform grids and outperforms graph convolutions on non-uniform data.
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.