There is a growing attention given to utilizing Lagrangian and Hamiltonian
mechanics with network training in order to incorporate physics into the
network. Most commonly, conservative systems are modeled, in which there are no
frictional losses, so the system may be run forward and backward in time
without requiring regularization. This work addresses systems in which the
reverse direction is ill-posed because of the dissipation that occurs in
forward evolution. The novelty is the use of Morse-Feshbach Lagrangian, which
models dissipative dynamics by doubling the number of dimensions of the system
in order to create a mirror latent representation that would counterbalance the
dissipation of the observable system, making it a conservative system, albeit
embedded in a larger space. We start with their formal approach by redefining a
new Dissipative Lagrangian, such that the unknown matrices in the
Euler-Lagrange's equations arise as partial derivatives of the Lagrangian with
respect to only the observables. We then train a network from simulated
training data for dissipative systems such as Fickian diffusion that arise in
materials sciences. It is shown by experiments that the systems can be evolved
in both forward and reverse directions without regularization beyond that
provided by the Morse-Feshbach Lagrangian. Experiments of dissipative systems,
such as Fickian diffusion, demonstrate the degree to which dynamics can be
reversed.

利用拉格朗日和哈密顿力学与网络训练来将物理学纳入网络中，以模拟具有耗散动力学的系统，本研究使用莫尔斯 - 费希巴赖格朗日方法模拟耗散系统，并通过网络训练数据表明该方法能够实现系统的正反向演化。

拉格朗日神经网络用于可逆耗散演化

Lagrangian Neural Networks for Reversible Dissipative Evolution

Forecasting of time-series data requires imposition of inductive biases to
obtain predictive extrapolation, and recent works have imposed
Hamiltonian/Lagrangian form to preserve structure for systems with reversible
dynamics. In this work we present a novel parameterization of dissipative
brackets from metriplectic dynamical systems appropriate for learning
irreversible dynamics with unknown a priori model form. The process learns
generalized Casimirs for energy and entropy guaranteed to be conserved and
nondecreasing, respectively. Furthermore, for the case of added thermal noise,
we guarantee exact preservation of a fluctuation-dissipation theorem, ensuring
thermodynamic consistency. We provide benchmarks for dissipative systems
demonstrating learned dynamics are more robust and generalize better than
either "black-box" or penalty-based approaches.

本文介绍了一种学习无可逆动力学的新方法，该方法利用 metriplectic 动力学系统的耗散括号的新参数化来学习泛化 Casimirs，此外，我们保证热噪声的情况下精确保存涨落耗散定理以保证热力学一致性。