用确定性方法高效学习复杂噪声下的动态系统，训练速度更快且支持并行计算。
Neural Kolmogorov Equations: Parallelizable Learning of Stochastic Dynamics under General Noise
- 基于柯尔莫哥洛夫前向方程，将随机微分方程转为概率密度演化建模
- 在耦合噪声和跳跃过程上表现优异，预测精度媲美传统方法
- 支持时间并行训练，大幅降低计算成本，适合高维动态系统建模
神经随机微分方程（Neural SDEs）已成为从数据中直接学习噪声动力学的强大工具，但现有方法多假设噪声解耦且连续，难以应对真实世界的复杂随机驱动，并常因自回归训练导致时间扩展性差。为此，我们提出神经柯尔莫哥洛夫方程（NKEs），基于柯尔莫哥洛夫前向方程（KFE）对神经SDE进行确定性、无限维重构，将学习问题从模拟单个随机轨迹转变为建模概率密度的演化。NKEs通过KFE的算子结构直接学习广义勒维型随机驱动，并利用拉格朗日伽辽金投影与算子分裂实现时间并行训练。我们在多个含耦合噪声和跳跃过程的基准测试中验证了其性能，结果表明NKEs具备灵活性，能准确恢复确定性与随机动力学，在预测精度上具有竞争力，同时显著提升训练效率。代码与预训练模型将公开发布。
原文摘要 · Abstract (English)
Neural stochastic differential equations (SDEs) have emerged as powerful tools for learning noisy or stochastic dynamics directly from data; however, existing approaches largely assume uncoupled and continuous noise, limiting their applicability to realistic stochastic drivers, and often scale poorly in time, requiring expensive autoregressive training. To address these limitations, we propose Neural Kolmogorov Equations (NKEs), a deterministic, infinite-dimensional reformulation of Neural SDEs based on the Kolmogorov Forward equation, transforming the learning problem from modelling individual stochastic trajectories to modelling the evolution of probability densities. NKEs learn general Lévy-type stochastic forcing directly through the operator structure of the KFE, and enable parallel-in-time training via a Lagrangian Galerkin projection and operator splitting. We evaluate NKEs on several stochastic benchmarks, including systems with coupled noise and jump processes, and verify that NKEs provide flexible models that accurately recover deterministic and stochastic dynamics with competitive predictive accuracy and improved training efficiency. Code and pretrained models will be released.
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