Physical theories grounded in mathematical symmetries are an essential
component of our understanding of a wide range of properties of the universe.
Similarly, in the domain of machine learning, an awareness of symmetries such
as rotation or permutation invariance has driven impressive performance
breakthroughs in computer vision, natural language processing, and other
important applications. In this report, we argue that both the physics
community and the broader machine learning community have much to understand
and potentially to gain from a deeper investment in research concerning
symmetry group equivariant machine learning architectures. For some
applications, the introduction of symmetries into the fundamental structural
design can yield models that are more economical (i.e. contain fewer, but more
expressive, learned parameters), interpretable (i.e. more explainable or
directly mappable to physical quantities), and/or trainable (i.e. more
efficient in both data and computational requirements). We discuss various
figures of merit for evaluating these models as well as some potential benefits
and limitations of these methods for a variety of physics applications.
Research and investment into these approaches will lay the foundation for
future architectures that are potentially more robust under new computational
paradigms and will provide a richer description of the physical systems to
which they are applied.

本文探讨了在物理学和机器学习领域中，研究对称群等变机器学习结构所带来的深度投资和收益，并讨论了应用这些方法的潜在益处和限制以及对不同物理应用的各种评估指标。

物理学的对称群等变架构

Symmetry Group Equivariant Architectures for Physics

Neural networks are a central technique in machine learning. Recent years
have seen a wave of interest in applying neural networks to physical systems
for which the governing dynamics are known and expressed through differential
equations. Two fundamental challenges facing the development of neural networks
in physics applications is their lack of interpretability and their
physics-agnostic design. The focus of the present work is to embed physical
constraints into the structure of the neural network to address the second
fundamental challenge. By constraining tunable parameters (such as weights and
biases) and adding special layers to the network, the desired constraints are
guaranteed to be satisfied without the need for explicit regularization terms.
This is demonstrated on upervised and unsupervised networks for two basic
symmetries: even/odd symmetry of a function and energy conservation. In the
supervised case, the network with embedded constraints is shown to perform well
on regression problems while simultaneously obeying the desired constraints
whereas a traditional network fits the data but violates the underlying
constraints. Finally, a new unsupervised neural network is proposed that
guarantees energy conservation through an embedded symplectic structure. The
symplectic neural network is used to solve a system of energy-conserving
differential equations and out-performs an unsupervised, non-symplectic neural
network.

本文的重点是将物理约束嵌入神经网络的结构中，以解决神经网络在物理应用中缺乏可解释性和物理不可知设计的问题，通过限制可调参数并添加特殊层，保证所需约束不需要显式正则化项即可满足，为解决函数的奇偶对称性和能量守恒问题提出了监督和非监督网络，并提出了一种嵌入所谓辛结构的无监督神经网络解决系统的守恒微分方程，表现出比非辛神经网络更好的性能。