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.

翻译：当前，越来越多的研究关注于将拉格朗日力学和哈密顿力学与网络训练相结合，以便将物理学原理融入网络之中。最常见的是对保守系统进行建模，这类系统不存在摩擦损耗，因此可以在时间上向前和向后运行，而无需进行正则化。本研究针对的是那些由于正向演化中存在耗散而导致反向演化不适定的系统。其新颖之处在于使用了莫尔斯-费什巴赫拉格朗日量，该方法通过将系统维度加倍来建模耗散动力学，从而创建一个镜像潜在表示，用以抵消可观测系统的耗散，使其成为一个保守系统，尽管是嵌入在一个更大的空间中。我们首先从他们的形式化方法出发，重新定义了一个新的耗散拉格朗日量，使得欧拉-拉格朗日方程中的未知矩阵仅作为拉格朗日量对可观测量的偏导数出现。随后，我们利用模拟训练数据训练了一个网络，用于处理材料科学中出现的耗散系统，如菲克扩散。实验表明，系统可以在正向和反向两个方向上演化，而无需超出莫尔斯-费什巴赫拉格朗日量所提供的正则化。对耗散系统（如菲克扩散）的实验，展示了动力学可逆的程度。