论文标题
使用数据调整内核的随机微分方程的一声学习
One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels
论文作者
论文摘要
我们考虑从一个样本轨迹中学习$ dx_t = f（x_t）dt+σ（x_t）dt+σ（x_t）dt+σ的随机微分方程的问题。这个问题比学习确定性动力学系统更具挑战性，因为一个样本轨迹仅提供有关未知功能$ f $，$σ$和随机过程$ dw_t $的间接信息，分别代表漂移，扩散和随机强迫术语。我们提出了一种结合计算图完成和通过新的交叉验证变体所学的数据改编的内核的方法。我们的方法可以分解如下：（1）表示时间添加映射$ x_t \ rightarrow x_ {t+dt} $作为计算图，其中$ f $，$σ$和$ dw_t $显示为未知功能和随机变量。 （2）通过在未知功能上使用高斯过程（GP）先验的最大后验估计（给定数据）来完成图（近似未知的函数和随机变量）。 （3）从具有随机交叉验证的数据中学习GP先验的协方差函数（内核）。数值实验说明了我们方法的功效，鲁棒性和范围。
We consider the problem of learning Stochastic Differential Equations of the form $dX_t = f(X_t)dt+σ(X_t)dW_t $ from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions $f$, $σ$, and stochastic process $dW_t$ representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map $X_t \rightarrow X_{t+dt}$ as a Computational Graph in which $f$, $σ$ and $dW_t$ appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.
