Modeling the complex three-dimensional (3D) dynamics of relational systems is an important problem in the natural sciences, with applications ranging from molecular simulations to particle mechanics. Machine learning methods have achieved good success by learning graph neural networks to model spatial interactions. However, these approaches do not faithfully capture temporal correlations since they only model next-step predictions. In this work, we propose Equivariant Graph Neural Operator (EGNO), a novel and principled method that directly models dynamics as trajectories instead of just next-step prediction. Different from existing methods, EGNO explicitly learns the temporal evolution of 3D dynamics where we formulate the dynamics as a function over time and learn neural operators to approximate it. To capture the temporal correlations while keeping the intrinsic SE(3)-equivariance, we develop equivariant temporal convolutions parameterized in the Fourier space and build EGNO by stacking the Fourier layers over equivariant networks. EGNO is the first operator learning framework that is capable of modeling solution dynamics functions over time while retaining 3D equivariance. Comprehensive experiments in multiple domains, including particle simulations, human motion capture, and molecular dynamics, demonstrate the significantly superior performance of EGNO against existing methods, thanks to the equivariant temporal modeling.

翻译：在自然科学领域，对关系系统的复杂三维动力学建模是一项重要问题，其应用涵盖从分子模拟到粒子力学等领域。机器学习方法通过学习图神经网络建模空间相互作用取得了良好成效。然而，这些方法仅能实现单步预测，未能精确捕捉时间相关性。本研究提出等变图神经算子——一种新颖且严谨的方法，该方法直接以轨迹形式建模动力学过程，而非仅进行单步预测。与现有方法不同，EGNO显式学习三维动力学的时间演化过程，将动力学定义为时间函数并通过学习神经算子对其进行近似。为在保持固有SE(3)-等变性的同时捕捉时间相关性，我们开发了在傅里叶空间中参数化的等变时间卷积，并通过在等变网络上堆叠傅里叶层构建EGNO。EGNO是首个能够建模随时间演化的求解动力学函数且保持三维等变性的算子学习框架。在粒子模拟、人体运动捕捉和分子动力学等多个领域的综合实验表明，得益于等变时间建模能力，EGNO相较于现有方法展现出显著优越的性能。