QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compression

TL;DR

QuadConv introduces a quadrature-based convolution layer designed for non-uniform mesh data, enabling efficient deep learning on PDE data with improved performance over existing unstructured convolution methods.

Contribution

It presents a novel continuous convolution operator, QuadConv, optimized for non-uniform, mesh-based data, with an efficient implementation and demonstrated effectiveness in PDE data compression.

Findings

QuadConv matches standard convolution performance on uniform grid data.

QuadConv maintains accuracy on non-uniform data.

QuadConv outperforms graph convolution methods.

Abstract

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.