This work introduces E3x, a software package for building neural networks that are equivariant with respect to the Euclidean group $\mathrm{E}(3)$, consisting of translations, rotations, and reflections of three-dimensional space. Compared to ordinary neural networks, $\mathrm{E}(3)$-equivariant models promise benefits whenever input and/or output data are quantities associated with three-dimensional objects. This is because the numeric values of such quantities (e.g. positions) typically depend on the chosen coordinate system. Under transformations of the reference frame, the values change predictably, but the underlying rules can be difficult to learn for ordinary machine learning models. With built-in $\mathrm{E}(3)$-equivariance, neural networks are guaranteed to satisfy the relevant transformation rules exactly, resulting in superior data efficiency and accuracy. The code for E3x is available from https://github.com/google-research/e3x, detailed documentation and usage examples can be found on https://e3x.readthedocs.io.

翻译：本文介绍了E3x，一个用于构建对三维空间中的平移、旋转和反射构成的欧几里得群 $\mathrm{E}(3)$ 具有等变性的神经网络的软件包。与普通神经网络相比，$\mathrm{E}(3)$-等变模型在输入和/或输出数据是与三维物体相关的量时具有优势。这是因为这类量（例如位置）的数值通常依赖于所选的坐标系。在参考系变换下，这些数值会以可预测的方式变化，但普通机器学习模型难以学习其背后的规则。通过内置的 $\mathrm{E}(3)$-等变性，神经网络能够保证精确满足相关变换规则，从而提升数据效率和准确性。E3x的代码可从 https://github.com/google-research/e3x 获取，详细文档和使用示例见 https://e3x.readthedocs.io。