Gaussian Process Latent Force Models for Learning and Stochastic Control of Physical Systems

Abstract--This paper is concerned with estimation and stochastic control in physical systems which contain unknown input signals or forces. These unknown signals are modeled as Gaussian processes (GP) in the sense that GP models are used in machine learning. The resulting latent force models (LFMs) can be seen as hybrid models that contain a first-principles physical model part and a nonparametric GP model part. The aim of this paper is to collect and extend the statistical inference and learning methods for this kind of models, provide new theoretical results for the models, and to extend the methodology and theory to stochastic control of LFMs. The generalizations of this kind of models to arbitrary differential equations are called latent force models (LFM) [2]-[6] in machine learning literature. In addition to learning problem on the LFMs, we also consider the problem of controlling the LFM using the control functionc(t) . In particular, we consider the problem of optimal stochastic control design for LFMs. The present problem is also closely related to so called input estimation problem that has previously been addressed in target tracking literature (e.g. Simo S arkk a is with the Department of Electrical Engineering and Automation (EEA), Aalto University, Rakentajanaukio 2c, 02150 Espoo, Finland (simo.sarkka@aalto.fi). The difference is that here is no concept of time in this equation, nor a possibility for controlling the equation. A. General problem formulation The models considered in this article can be seen to belong to the following three classes: 1) Basic latent force models which are ordinary differential equations (ODEs) driven by Gaussian input processes u (t) and control inputsc(t) . X, MONTH 20XX 2 2) We also consider are dynamic partial and pseudo differential equation (PDE) based models that can generally be written in form L f (x,t) u (x,t) c(x,t), (7) where L is a linear operator in space and time. The input Gaussian processu (x,t) and control inputc(x,t) are also space-time processes. Typically, the operator has the form L A m d m dt m ··· A 1 d dt A 0, (8) where A 0,...,A m are some spatial partial differential or pseudo-differential operators. This kind of models can often be also written in form of spatiotemporal state-space models f (x,t) t A f f (x,t) B f u (x,t) M f c (x,t), (9) which again is strictly more general than the model (8). For this kind of models there is no control problem per se, because there is no time dependence. These models do not naturally allow for a state-space representation either.