Spherical convolutions and their application in molecular modelling

Convolutional neural networks are increasingly used outside the domain of image analysis, in particular in various areas of the Natural Sciences concerned with spatial data. Such networks often work out-of-the box, and in some cases entire model architectures from image analysis can be carried over to other problem domains almost unaltered. Unfortunately, this convenience does not trivially extend to data in non-euclidean spaces, such as spherical data. In this paper, we address the challenges that arise in this setting, in particular the lack of translational equivariance associated with using a grid based on uniform spacing in spherical coordinates. We present a definition of a spherical convolution that overcomes these issues, and extend our discussion to include scenarios of spherical volumes, with several strategies for parameterizing the radial dimension. As a proof of concept, we conclude with an assessment of the performance of spherical convolutions in the context of molecular modelling, by considering structural environments within proteins. We show that the model is capable of learning non-trivial functions in these molecular environments, and despite the lack of any domain specific feature-engineering, we demonstrate performance comparable to state-of-the-art methods in the field, which build on decades of domain-specific knowledge.