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. Our code is available at https://github.com/MinkaiXu/egno.

翻译：建模关系系统的复杂三维动力学是自然科学中的一个重要问题，其应用范围涵盖分子模拟至粒子力学。机器学习方法通过学习图神经网络来建模空间相互作用已取得良好成效。然而，这些方法仅建模下一步预测，未能忠实捕捉时间相关性。本工作中，我们提出等变图神经算子（EGNO），这是一种新颖且原理性明确的方法，直接将动力学建模为轨迹而非仅进行下一步预测。与现有方法不同，EGNO显式学习三维动力学的时间演化，将动力学表述为时间的函数并学习神经算子进行近似。为在保持内在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是首个能够在保持三维等变性的同时对时间维度上的解动力学函数进行建模的算子学习框架。在粒子模拟、人体运动捕捉及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，EGNO相比现有方法具有显著优越的性能。代码发布于 https://github.com/MinkaiXu/egno。