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

TL;DR

QuadConv introduces a quadrature-based convolution that approximates the continuous convolution on non-uniform meshes by learning a continuous kernel $g$ and evaluating it via quadrature at arbitrary points. The method supports multi-channel data, uses a bump-based compact kernel with a learned map $H(\cdot; \theta)$, and optionally learns quadrature weights $\rho_i$, with a caching strategy for the input mesh to enable efficient computation. Empirical results on PDE data compression show that QuadConv can match standard convolutions on uniform grids and outperform unstructured alternatives on non-uniform meshes, across multiple datasets (uniform ignition, non-uniform ignition, non-uniform flow). The work demonstrates practical implementation details, including index-map construction, sparsity exploitation, and complexity considerations, and discusses future directions such as multi-resolution training and online learning for in situ compression.

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.

Figures (18)

• Figure 1: Examples of non-uniform data.
• Figure 2: Comparison of continuous and discrete convolution.
• Figure 3: Comparison of discrete convolution and quadrature-based convolution on a non-uniformly sampled signal.
• Figure 4: Comparison of discrete convolution and QuadConv.
• Figure 5: QuadConv computation. The output value at index $j=2$ depends only sparsely on the input values, e.g., there is no dependence on the input value at index $i=2$.