Efficient Long-Range Convolutions for Point Clouds

The efficient treatment of long-range interactions for point clouds is a challenging problem in many scientific machine learning applications. To extract global information, one usually needs a large window size, a large number of layers, and/or a large number of channels. This can often significantly increase the computational cost. In this work, we present a novel neural network layer that directly incorporates long-range information for a point cloud. This layer, dubbed the long-range convolutional (LRC)-layer, leverages the convolutional theorem coupled with the non-uniform Fourier transform. In a nutshell, the LRC-layer mollifies the point cloud to an adequately sized regular grid, computes its Fourier transform, multiplies the result by a set of trainable Fourier multipliers, computes the inverse Fourier transform, and finally interpolates the result back to the point cloud. The resulting global all-to-all convolution operation can be performed in nearly-linear time asymptotically with respect to the number of input points. The LRC-layer is a particularly powerful tool when combined with local convolution as together they offer efficient and seamless treatment of both short and long range interactions. We showcase this framework by introducing a neural network architecture that combines LRC-layers with short-range convolutional layers to accurately learn the energy and force associated with a $N$-body potential. We also exploit the induced two-level decomposition and propose an efficient strategy to train the combined architecture with a reduced number of samples.