We develop a method to learn physical systems from data that employs
feedforward neural networks and whose predictions comply with the first and
second principles of thermodynamics. The method employs a minimum amount of
data by enforcing the metriplectic structure of dissipative Hamiltonian systems
in the form of the so-called General Equation for the Non-Equilibrium
Reversible-Irreversible Coupling, GENERIC [M. Grmela and H.C Oettinger (1997).
Dynamics and thermodynamics of complex fluids. I. Development of a general
formalism. Phys. Rev. E. 56 (6): 6620-6632]. The method does not need to
enforce any kind of balance equation, and thus no previous knowledge on the
nature of the system is needed. Conservation of energy and dissipation of
entropy in the prediction of previously unseen situations arise as a natural
by-product of the structure of the method. Examples of the performance of the
method are shown that include conservative as well as dissipative systems,
discrete as well as continuous ones.

利用前馈神经网络和热力学第一和第二定律，通过从数据中学习物理系统的方法，以及使用所谓的非平衡可逆耦合通用方程 (GENERIC) 的度规辛结构，可以最小化地使用数据。该方法不需要强制任何平衡方程式，因此不需要关于系统性质的任何先前知识。能量守恒和熵耗散是该方法结构的自然副产品。展示了该方法的性能示例，包括守恒和耗散系统以及离散和连续系统。

结构保持神经网络

Structure-preserving neural networks

Arguably, the two most popular accelerated or momentum-based optimization
methods in machine learning are Nesterov's accelerated gradient and Polyaks's
heavy ball, both corresponding to different discretizations of a particular
second order differential equation with friction. Such connections with
continuous-time dynamical systems have been instrumental in demystifying
acceleration phenomena in optimization. Here we study structure-preserving
discretizations for a certain class of dissipative (conformal) Hamiltonian
systems, allowing us to analyze the symplectic structure of both Nesterov and
heavy ball, besides providing several new insights into these methods.
Moreover, we propose a new algorithm based on a dissipative relativistic system
that normalizes the momentum and may result in more stable/faster optimization.
Importantly, such a method generalizes both Nesterov and heavy ball, each being
recovered as distinct limiting cases, and has potential advantages at no
additional cost.

本文研究了一类耗散哈密顿系统的结构保存离散化方法及其在机器学习中的应用，包括流行的加速优化算法 Nesterov 及 Polyak's heavy ball 的初步分析和新洞见，同时提出了一种基于耗散相对论系统的新算法可应用于加速优化但规避额外成本