发表机构
EPFL; Imperial College（洛桑联邦理工学院; 帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对神经随机微分方程现有方法的局限性，提出神经柯尔莫哥洛夫方程，基于柯尔莫哥洛夫前向方程重构，通过其算子结构学习随机强迫，利用拉格朗日伽辽金投影和算子分裂实现并行训练，在多基准测试中表现良好。
AI 中文摘要
神经随机微分方程（SDEs）已成为直接从数据中学习噪声或随机动力学的强大工具，但现有方法大多假设噪声非耦合且连续，限制了其对现实随机驱动的适用性，且时间尺度不佳，需要昂贵的自回归训练。为解决这些限制，我们提出神经柯尔莫哥洛夫方程（NKEs），它基于柯尔莫哥洛夫前向方程对神经SDEs进行确定性、无限维重构，将学习问题从建模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一般的 Lévy 型随机强迫，并通过拉格朗日伽辽金投影和算子分裂实现并行时间训练。我们在几个随机基准上评估了NKEs，包括具有耦合噪声和跳跃过程的系统，并验证了NKEs提供了灵活的模型，能以有竞争力的预测精度和提高的训练效率准确恢复确定性和随机动力学。代码和预训练模型将发布。
英文摘要
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.
