Optimal control deals with optimization problems in which variables steer a
dynamical system, and its outcome contributes to the objective function. Two
classical approaches to solving these problems are Dynamic Programming and the
Pontryagin Maximum Principle. In both approaches, Hamiltonian equations offer
an interpretation of optimality through auxiliary variables known as costates.
However, Hamiltonian equations are rarely used due to their reliance on
forward-backward algorithms across the entire temporal domain. This paper
introduces a novel neural-based approach to optimal control, with the aim of
working forward-in-time. Neural networks are employed not only for implementing
state dynamics but also for estimating costate variables. The parameters of the
latter network are determined at each time step using a newly introduced local
policy referred to as the time-reversed generalized Riccati equation. This
policy is inspired by a result discussed in the Linear Quadratic (LQ) problem,
which we conjecture stabilizes state dynamics. We support this conjecture by
discussing experimental results from a range of optimal control case studies.

本文介绍了一种新颖的基于神经网络的最优控制方法，该方法旨在通过在时间上前进来实现最优控制。神经网络不仅用于实施状态动态，还用于估计共轭变量。通过引入一种新的局部策略，即时间反转的广义 Riccati 方程，来确定后一网络的参数。通过讨论一系列最优控制案例研究的实验结果，我们支持这个猜想，即这个策略能够稳定状态动态。

神经时间反演广义瑞克蒂方程

Neural Time-Reversed Generalized Riccati Equation

In many real-world settings, image observations of freely rotating 3D rigid
bodies, such as satellites, may be available when low-dimensional measurements
are not. However, the high-dimensionality of image data precludes the use of
classical estimation techniques to learn the dynamics and a lack of
interpretability reduces the usefulness of standard deep learning methods. In
this work, we present a physics-informed neural network model to estimate and
predict 3D rotational dynamics from image sequences. We achieve this using a
multi-stage prediction pipeline that maps individual images to a latent
representation homeomorphic to $\mathbf{SO}(3)$, computes angular velocities
from latent pairs, and predicts future latent states using the Hamiltonian
equations of motion with a learned representation of the Hamiltonian. We
demonstrate the efficacy of our approach on a new rotating rigid-body dataset
with sequences of rotating cubes and rectangular prisms with uniform and
non-uniform density.

本文提出一种物理知识指导的神经网络模型，能够从图像序列中估算预测旋转刚体的 3D 动力学，多阶段预测管道映射图像到与 SO (3) 同胚的潜在表示，通过学习得到的哈密顿表示，利用哈密顿运动方程预测未来的潜在状态。