The solution of problems in physics is often facilitated by a change of
variables. In this work we present neural transformations to learn symmetries
of Hamiltonian mechanical systems. Maintaining the Hamiltonian structure
requires novel network architectures that parametrize symplectic
transformations. We demonstrate the utility of these architectures by learning
the structure of integrable models. Our work exemplifies the adaptation of
neural transformations to a family constrained by more than the condition of
invertibility, which we expect to be a common feature of applications of these
methods.

本研究通过神经变换来学习哈密顿机械系统的对称性，需要新的网络体系结构来参数化辛变换，以维持哈密顿结构，并学习了可积模型的结构，这是神经变换适应一个受限于反演之外的家庭的典型示例。

学习古典可积系统的对称性

Learning Symmetries of Classical Integrable Systems

Understanding the relationship which integrable (solvable) models, all of
which possess very special symmetry properties, have with the generic
non-integrable models that are used to describe real experiments, which do not
have the symmetry properties, is one of the most fundamental open questions in
both statistical mechanics and quantum field theory. The importance of the
two-dimensional Ising model in a magnetic field is that it is the simplest
system where this relationship may be concretely studied. We here review the
advances made in this study, and concentrate on the magnetic susceptibility
which has revealed an unexpected natural boundary phenomenon. When this is
combined with the Fermionic representations of conformal characters, it is
suggested that the scaling theory, which smoothly connects the lattice with the
correlation length scale, may be incomplete for $H \neq 0$.

本文回顾了在研究具有特殊对称性的可积（可解决）模型与用于描述实际实验的无对称性的泛在非可积模型之间的关系中所取得的进展，并集中讨论了磁化率，在此基础上结合共形字符的费米表示，暗示了在 $H 
eq 0$ 时，用于将晶格与相关长度尺度相连接的比例理论可能是不完整的。