Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLPs provide a new class of activation functions for general use with steerable feature fields. We discuss ours and related work through the lens of equivariant non-linear convolutions, which further allows us to pin-point the successful components of SEGNNs: non-linear message aggregation improves upon classic linear (steerable) point convolutions; steerable messages improve upon recent equivariant graph networks that send invariant messages. We demonstrate the effectiveness of our method on several tasks in computational physics and chemistry and provide extensive ablation studies.

Normalising flows (NFs) for discrete data are challenging because parameterising bijective transformations of discrete variables requires predicting discrete/integer parameters. Having a neural network architecture predict discrete parameters takes a non-differentiable activation function (eg, the step function) which precludes gradient-based learning. To circumvent this non-differentiability, previous work has employed biased proxy gradients, such as the straight-through estimator. We present an unbiased alternative where rather than deterministically parameterising one transformation, we predict a distribution over latent transformations. With stochastic transformations, the marginal likelihood of the data is differentiable and gradient-based learning is possible via score function estimation. To test the viability of discrete-data NFs we investigate performance on binary MNIST. We observe great challenges with both deterministic proxy gradients and unbiased score function estimation. Whereas the former often fails to learn even a shallow transformation, the variance of the latter could not be sufficiently controlled to admit deeper NFs.