Incorporating Euclidean symmetries (e.g. rotation equivariance) as inductive
biases into graph neural networks has improved their generalization ability and
data efficiency in unbounded physical dynamics modeling. However, in various
scientific and engineering applications, the symmetries of dynamics are
frequently discrete due to the boundary conditions. Thus, existing GNNs either
overlook necessary symmetry, resulting in suboptimal representation ability, or
impose excessive equivariance, which fails to generalize to unobserved
symmetric dynamics. In this work, we propose a general Discrete Equivariant
Graph Neural Network (DEGNN) that guarantees equivariance to a given discrete
point group. Specifically, we show that such discrete equivariant message
passing could be constructed by transforming geometric features into
permutation-invariant embeddings. Through relaxing continuous equivariant
constraints, DEGNN can employ more geometric feature combinations to
approximate unobserved physical object interaction functions. Two
implementation approaches of DEGNN are proposed based on ranking or pooling
permutation-invariant functions. We apply DEGNN to various physical dynamics,
ranging from particle, molecular, crowd to vehicle dynamics. In twenty
scenarios, DEGNN significantly outperforms existing state-of-the-art
approaches. Moreover, we show that DEGNN is data efficient, learning with less
data, and can generalize across scenarios such as unobserved orientation.

本文提出了一种基于离散等变的图神经网络（DEGNN），通过将几何特征转化为排列不变嵌入，实现相应离散点群的等变消息传递，从而提高对未观测到的对称动力学的表示能力和泛化性能。在各种物理动力学中，DEGNN 相较于现有的方法具有显著的优越性，并可以以较少的数据进行学习，并且可以跨领域进行泛化，如未观测到的方向。

对等图神经网络在物理动力学学习中松弛连续约束

Relaxing Continuous Constraints of Equivariant Graph Neural Networks for  Physical Dynamics Learning

Identification and analysis of symmetrical patterns in the natural world have
led to significant discoveries across various scientific fields, such as the
formulation of gravitational laws in physics and advancements in the study of
chemical structures. In this paper, we focus on exploiting Euclidean symmetries
inherent in certain cooperative multi-agent reinforcement learning (MARL)
problems and prevalent in many applications. We begin by formally
characterizing a subclass of Markov games with a general notion of symmetries
that admits the existence of symmetric optimal values and policies. Motivated
by these properties, we design neural network architectures with symmetric
constraints embedded as an inductive bias for multi-agent actor-critic methods.
This inductive bias results in superior performance in various cooperative MARL
benchmarks and impressive generalization capabilities such as zero-shot
learning and transfer learning in unseen scenarios with repeated symmetric
patterns. The code is available at: this https URL

本文重点研究了自然界中对称模式的识别和分析，在物理学中导致了引力定律的制定和化学结构研究的进展。我们着眼于利用某些协同多智能体强化学习问题中固有的欧几里得对称性，该问题在许多应用中普遍存在。我们首先形式化地表征了一类具有对称最优值和策略存在性的马尔科夫博弈的子类。在这些属性的基础上，我们设计了具有对称约束的神经网络架构，作为多智能体演员 - 评论家方法的归纳偏见。这种归纳偏见在各种协同多智能体强化学习基准测试中表现出优越的性能，以及在具有重复对称模式的未见场景中进行的零样本学习和迁移学习等令人印象深刻的泛化能力。代码可在此 https URL 获取。