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。