  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.
