论文标题
识别潜在的随机微分方程
Identifying Latent Stochastic Differential Equations
论文作者
论文摘要
我们提出了一种从高维时间序列数据中学习潜在随机微分方程（SDE）的方法。鉴于从较低维的潜在未知ITô过程产生的高维时间序列，提出的方法通过一种自我监督的学习方法来学习从环境到潜在空间以及潜在的SDE系数的映射。使用变异自动编码器的框架，我们根据SDE溶液的Euler-Maruyama近似来考虑数据的条件生成模型。此外，我们对潜在变量模型的可识别性使用最新结果，以表明所提出的模型不仅可以恢复基础的SDE系数，还可以在无限数据的限制下恢复原始的潜在变量，最多可恢复等轴测图。我们通过几个模拟的视频处理任务来验证该方法，该任务已知基础SDE以及通过现实世界数据集验证。
We present a method for learning latent stochastic differential equations (SDEs) from high-dimensional time series data. Given a high-dimensional time series generated from a lower dimensional latent unknown Itô process, the proposed method learns the mapping from ambient to latent space, and the underlying SDE coefficients, through a self-supervised learning approach. Using the framework of variational autoencoders, we consider a conditional generative model for the data based on the Euler-Maruyama approximation of SDE solutions. Furthermore, we use recent results on identifiability of latent variable models to show that the proposed model can recover not only the underlying SDE coefficients, but also the original latent variables, up to an isometry, in the limit of infinite data. We validate the method through several simulated video processing tasks, where the underlying SDE is known, and through real world datasets.
