Data in many applications follows systems of Ordinary Differential Equations (ODEs). This paper presents a novel algorithmic and symbolic construction for covariance functions of Gaussian Processes (GPs) with realizations strictly following a system of linear homogeneous ODEs with constant coefficients, which we call LODE-GPs. Introducing this strong inductive bias into a GP improves modelling of such data. Using smith normal form algorithms, a symbolic technique, we overcome two current restrictions in the state of the art: (1) the need for certain uniqueness conditions in the set of solutions, typically assumed in classical ODE solvers and their probabilistic counterparts, and (2) the restriction to controllable systems, typically assumed when encoding differential equations in covariance functions. We show the effectiveness of LODE-GPs in a number of experiments, for example learning physically interpretable parameters by maximizing the likelihood.
翻译:许多应用中的数据都遵循普通差异方程式(ODEs)的系统。本文为Gaussian Processes(GPs)的共变量功能提供了一种新型的算法和象征性的构建,其实现严格遵循一个具有恒定系数的线性同质式数码系统,我们称之为LODE-GPs。将这种强烈的感应偏差引入GP的模型改进了这些数据的模型。我们使用机械式的普通形式方程式算法(一种象征性技术),克服了目前艺术状态中的两种限制:(1) 需要一套解决方案中的某些独特性条件,通常是古典的ODE解答器及其概率对应方所假定的;(2) 对可控系统的限制,通常是在共变函数中的编码差异方程式时假设的。我们在若干实验中展示了LODE-GPs的有效性,例如通过尽可能增加可能性来学习物理解释参数。