There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincar\'e groups, at any dimensionality $d$. The key observation is that nonlinear O($d$)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars -- scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.

翻译：近年来，在设计尊重物理定律基本对称性和坐标自由度的神经网络方面取得了巨大进展。一些框架利用不可约表示，一些使用高阶张量对象，还有一些应用对称性约束。不同的物理定律遵循不同的基本对称性组合，但经典物理的绝大部分（可能全部）对于平移、旋转、反射（宇称）、加速（相对性）和置换是等变的。在此，我们展示了一种简单的方法，可以参数化在任意维度$d$下对于这些对称性，或在欧几里得群、洛伦兹群和庞加莱群下普适逼近的多项式函数。关键发现是，非线性O($d$)等变（及相关群等变）函数可以通过一组轻量级标量——标量、矢量和张量输入的标量积和标量缩并——来普适地表达。我们用数值示例补充了理论，表明基于标量的方法简单、高效且可扩展。