Steerable Partial Differential Operators for Equivariant Neural Networks
This work bridges physics and deep learning by developing a theoretical framework for equivariant PDOs, offering incremental improvements in equivariant neural networks for applications requiring symmetry-aware models.
The paper tackles the problem of characterizing equivariant partial differential operators (PDOs) for arbitrary symmetry groups in deep learning, deriving a steerability constraint and solving it for several groups, with results showing competitive performance when used as drop-in replacements for convolutional layers in benchmarks.
Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a $G$-steerability constraint that completely characterizes when a PDO between feature vector fields is equivariant, for arbitrary symmetry groups $G$. We then fully solve this constraint for several important groups. We use our solutions as equivariant drop-in replacements for convolutional layers and benchmark them in that role. Finally, we develop a framework for equivariant maps based on Schwartz distributions that unifies classical convolutions and differential operators and gives insight about the relation between the two.