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.

研究在相等变深度学习和物理领域中共享一致的数学概念，并通过极化微分算子和基于调和分析的方法提出了一个新的框架，使我们能够更好地将基于显式对称性应变的捕捉无论是在深度学习还是物理学中都可以更精确地进行表达。

可变等变神经网络的可定向偏微分算子

Steerable Partial Differential Operators for Equivariant Neural Networks

We describe generalizations of the universal approximation theorem for neural
networks to maps invariant or equivariant with respect to linear
representations of groups. Our goal is to establish network-like computational
models that are both invariant/equivariant and provably complete in the sense
of their ability to approximate any continuous invariant/equivariant map. Our
contribution is three-fold. First, in the general case of compact groups we
propose a construction of a complete invariant/equivariant network using an
intermediate polynomial layer. We invoke classical theorems of Hilbert and Weyl
to justify and simplify this construction; in particular, we describe an
explicit complete ansatz for approximation of permutation-invariant maps.
Second, we consider groups of translations and prove several versions of the
universal approximation theorem for convolutional networks in the limit of
continuous signals on euclidean spaces. Finally, we consider 2D signal
transformations equivariant with respect to the group SE(2) of rigid euclidean
motions. In this case we introduce the "charge--conserving convnet" -- a
convnet-like computational model based on the decomposition of the feature
space into isotypic representations of SO(2). We prove this model to be a
universal approximator for continuous SE(2)--equivariant signal
transformations.

我们将神经网络的普适逼近定理推广到对于线性表示组不变或等变的映射，以建立一种像网络一样的计算模型，能够在能够逼近任何连续不变 / 等变映射的同时保持不变 / 等变。我们提出了完备的不变 / 等变网络的构造，通过引入中间多项式层，通过 Hilbert 和 Weyl 的定理证明了我们的构造方法。我们提出了适用于 SE（2）群的 “电荷守恒卷积” 模型，并证明其是连续 SE（2）等变信号变换的通用逼近器。