Neural networks (NNs) that exploit strong inductive biases based on physical laws and symmetries have shown remarkable success in learning the dynamics of physical systems directly from their trajectory. However, these works focus only on the systems that follow deterministic dynamics, for instance, Newtonian or Hamiltonian dynamics. Here, we propose a framework, namely Brownian graph neural networks (BROGNET), combining stochastic differential equations (SDEs) and GNNs to learn Brownian dynamics directly from the trajectory. We theoretically show that BROGNET conserves the linear momentum of the system, which in turn, provides superior performance on learning dynamics as revealed empirically. We demonstrate this approach on several systems, namely, linear spring, linear spring with binary particle types, and non-linear spring systems, all following Brownian dynamics at finite temperatures. We show that BROGNET significantly outperforms proposed baselines across all the benchmarked Brownian systems. In addition, we demonstrate zero-shot generalizability of BROGNET to simulate unseen system sizes that are two orders of magnitude larger and to different temperatures than those used during training. Altogether, our study contributes to advancing the understanding of the intricate dynamics of Brownian motion and demonstrates the effectiveness of graph neural networks in modeling such complex systems.

翻译：神经网络（NNs）利用基于物理定律和对称性的强先验偏置，在直接从轨迹学习物理系统动力学方面取得了显著成功。然而，这些工作仅关注遵循确定性动力学的系统，例如牛顿力学或哈密顿动力学。本文提出一种名为布朗图神经网络（BROGNET）的框架，该框架结合随机微分方程（SDEs）与图神经网络（GNNs），直接从轨迹中学习布朗动力学。我们从理论上证明，BROGNET能够守恒系统的线性动量，进而如实证结果所示，在动力学学习任务中展现出卓越性能。我们在线性弹簧、含二元粒子类型的线性弹簧以及非线性弹簧等多个系统上验证了该方法，这些系统均在有限温度下遵循布朗动力学。研究表明，在所有基准布朗系统中，BROGNET显著优于提出的基线方法。此外，我们还展示了BROGNET的零样本泛化能力：它能模拟训练中未见过的、规模大两个数量级的系统，以及不同温度下的系统。总体而言，本研究深化了对布朗运动复杂动力学的理解，并证明了图神经网络在建模此类复杂系统中的有效性。