This work proposes an approach for latent-dynamics learning that exactly
enforces physical conservation laws. The method comprises two steps. First, the
method computes a low-dimensional embedding of the high-dimensional
dynamical-system state using deep convolutional autoencoders. This defines a
low-dimensional nonlinear manifold on which the state is subsequently enforced
to evolve. Second, the method defines a latent-dynamics model that associates
with the solution to a constrained optimization problem. Here, the objective
function is defined as the sum of squares of conservation-law violations over
control volumes within a finite-volume discretization of the problem; nonlinear
equality constraints explicitly enforce conservation over prescribed subdomains
of the problem. Under modest conditions, the resulting dynamics model
guarantees that the time-evolution of the latent state exactly satisfies
conservation laws over the prescribed subdomains.

本文提出了一种潜在动态学习的方法，该方法通过深度卷积自编码器计算高维动力系统状态的低维嵌入，并定义了一个低维的非线性流形使站在其上的状态被要求进行演化，在此基础上定义了潜在动力学模型，该模型的目标函数为有限体离散化中控制体内守恒定律违规平方和加上非线性等式约束，该模型能确保在预定子域上时间演化的潜在状态严格满足守恒定律。

Deep Conservation：一种隐式动力学模型，实现物理守恒定律的精确满足

Deep Conservation: A latent-dynamics model for exact satisfaction of  physical conservation laws

We introduce a conditional generative model for learning to disentangle the
hidden factors of variation within a set of labeled observations, and separate
them into complementary codes. One code summarizes the specified factors of
variation associated with the labels. The other summarizes the remaining
unspecified variability. During training, the only available source of
supervision comes from our ability to distinguish among different observations
belonging to the same class. Examples of such observations include images of a
set of labeled objects captured at different viewpoints, or recordings of set
of speakers dictating multiple phrases. In both instances, the intra-class
diversity is the source of the unspecified factors of variation: each object is
observed at multiple viewpoints, and each speaker dictates multiple phrases.
Learning to disentangle the specified factors from the unspecified ones becomes
easier when strong supervision is possible. Suppose that during training, we
have access to pairs of images, where each pair shows two different objects
captured from the same viewpoint. This source of alignment allows us to solve
our task using existing methods. However, labels for the unspecified factors
are usually unavailable in realistic scenarios where data acquisition is not
strictly controlled. We address the problem of disentanglement in this more
general setting by combining deep convolutional autoencoders with a form of
adversarial training. Both factors of variation are implicitly captured in the
organization of the learned embedding space, and can be used for solving
single-image analogies. Experimental results on synthetic and real datasets
show that the proposed method is capable of generalizing to unseen classes and
intra-class variabilities.

我们提出了一种有条件的生成模型，用于学习将标记的观察结果的隐藏变化因素分离并分解成互补代码，实验结果表明该方法能够推广到未见过的类别和内类别变异。