  Many machine learning tasks involve learning functions that are known to be
invariant or equivariant to certain symmetries of the input data. However, it
is often challenging to design neural network architectures that respect these
symmetries while being expressive and computationally efficient. For example,
Euclidean motion invariant/equivariant graph or point cloud neural networks. We
introduce Frame Averaging (FA), a general purpose and systematic framework for
adapting known (backbone) architectures to become invariant or equivariant to
new symmetry types. Our framework builds on the well known group averaging
operator that guarantees invariance or equivariance but is intractable. In
contrast, we observe that for many important classes of symmetries, this
operator can be replaced with an averaging operator over a small subset of the
group elements, called a frame. We show that averaging over a frame guarantees
exact invariance or equivariance while often being much simpler to compute than
averaging over the entire group. Furthermore, we prove that FA-based models
have maximal expressive power in a broad setting and in general preserve the
expressive power of their backbone architectures. Using frame averaging, we
propose a new class of universal Graph Neural Networks (GNNs), universal
Euclidean motion invariant point cloud networks, and Euclidean motion invariant
Message Passing (MP) GNNs. We demonstrate the practical effectiveness of FA on
several applications including point cloud normal estimation, beyond $2$-WL
graph separation, and $n$-body dynamics prediction, achieving state-of-the-art
results in all of these benchmarks.
