Neural networks are emerging as a tool for scalable data-driven simulation of high-dimensional dynamical systems, especially in settings where numerical methods are infeasible or computationally expensive. Notably, it has been shown that incorporating domain symmetries in deterministic neural simulators can substantially improve their accuracy, sample efficiency, and parameter efficiency. However, to incorporate symmetries in probabilistic neural simulators that can simulate stochastic phenomena, we need a model that produces equivariant distributions over trajectories, rather than equivariant function approximations. In this paper, we propose Equivariant Probabilistic Neural Simulation (EPNS), a framework for autoregressive probabilistic modeling of equivariant distributions over system evolutions. We use EPNS to design models for a stochastic n-body system and stochastic cellular dynamics. Our results show that EPNS considerably outperforms existing neural network-based methods for probabilistic simulation. More specifically, we demonstrate that incorporating equivariance in EPNS improves simulation quality, data efficiency, rollout stability, and uncertainty quantification. We conclude that EPNS is a promising method for efficient and effective data-driven probabilistic simulation in a diverse range of domains.

翻译：神经网络正逐渐成为高维动力系统可扩展数据驱动模拟的工具，尤其在数值方法不可行或计算成本高昂的场景中。值得注意的是，在确定性神经模拟器中融入领域对称性可显著提升其准确性、样本效率和参数效率。然而，为在可模拟随机现象的随机神经模拟器中融入对称性，需要一种能生成轨迹上等变分布而非等变函数近似的模型。本文提出等变概率神经模拟（EPNS）框架，用于系统演化过程中等变分布的自回归概率建模。我们利用EPNS设计了随机n体系统和随机细胞动力学的模型。实验结果表明，EPNS在随机模拟中显著优于现有基于神经网络的方法。具体而言，我们证明EPNS中融入等变性可改善模拟质量、数据效率、滚动稳定性及不确定性量化。我们得出结论：EPNS是跨多领域实现高效数据驱动随机模拟的极具前景的方法。