We propose a new family of neural networks to predict the behaviors of
physical systems by learning their underpinning constraints. A neural
projection operator lies at the heart of our approach, composed of a
lightweight network with an embedded recursive architecture that interactively
enforces learned underpinning constraints and predicts the various governed
behaviors of different physical systems. Our neural projection operator is
motivated by the position-based dynamics model that has been used widely in
game and visual effects industries to unify the various fast physics
simulators. Our method can automatically and effectively uncover a broad range
of constraints from observation point data, such as length, angle, bending,
collision, boundary effects, and their arbitrary combinations, without any
connectivity priors. We provide a multi-group point representation in
conjunction with a configurable network connection mechanism to incorporate
prior inputs for processing complex physical systems. We demonstrated the
efficacy of our approach by learning a set of challenging physical systems all
in a unified and simple fashion including: rigid bodies with complex
geometries, ropes with varying length and bending, articulated soft and rigid
bodies, and multi-object collisions with complex boundaries.

我们提出了一种新的神经网络家族，通过学习基础约束条件来预测物理系统的行为，其中我们的神经投影算子是核心，可以自动有效地发现各种约束条件，并预测不同物理系统的管辖行为。

用神经投影学习物理约束

Learning Physical Constraints with Neural Projections

We cast policy gradient methods as the repeated application of two operators:
a policy improvement operator $\mathcal{I}$, which maps any policy $\pi$ to a
better one $\mathcal{I}\pi$, and a projection operator $\mathcal{P}$, which
finds the best approximation of $\mathcal{I}\pi$ in the set of realizable
policies. We use this framework to introduce operator-based versions of
traditional policy gradient methods such as REINFORCE and PPO, which leads to a
better understanding of their original counterparts. We also use the
understanding we develop of the role of $\mathcal{I}$ and $\mathcal{P}$ to
propose a new global lower bound of the expected return. This new perspective
allows us to further bridge the gap between policy-based and value-based
methods, showing how REINFORCE and the Bellman optimality operator, for
example, can be seen as two sides of the same coin.

本文通过引入操作符的概念，将传统的强化学习算法中的策略梯度方法如 REINFORCE 和 PPO 等转化成了操作符形式，从而更好地理解它们的原理，同时通过引入新的全局下限，进一步弥合了基于策略和基于价值的方法之间的差距，将 REINFORCE 算法和贝尔曼最优化操作符视为同一概念的两个方面。

策略梯度方法的操作符视角

An operator view of policy gradient methods

We propose a scheme to restore spatial symmetry of Hamiltonian in the
variational-quantum-eigensolver (VQE) algorithm for which the quantum circuit
structures used usually break the Hamiltonian symmetry. The symmetry-adapted
VQE scheme introduced here simply applies the projection operator, which is
Hermitian but not unitary, to restore the spatial symmetry in a desired
irreducible representation of the spatial group. The entanglement of a quantum
state is still represented in a quantum circuit but the nonunitarity of the
projection operator is treated classically as postprocessing in the VQE
framework. By numerical simulations for a spin-$1/2$ Heisenberg model on a
one-dimensional ring, we demonstrate that the symmetry-adapted VQE scheme with
a shallower quantum circuit can achieve significant improvement in terms of the
fidelity of the ground state and has a great advantage in terms of the
ground-state energy with decent accuracy, as compared to the
non-symmetry-adapted VQE scheme. We also demonstrate that the present scheme
can approximate low-lying excited states that can be specified by symmetry
sectors, using the same circuit structure for the ground-state calculation.

本文介绍了一种恢复 VQE 算法中 Hamiltonian 对称性的算法方案，该方案应用投影算符来实现空间对称性的恢复，具有量子电路与非幺正投影算符的组合优势，能够在较浅的量子电路中实现显著的基态精度提升，并近似计算对称性限制下的激发态能量。