State estimation of a physical system with unknown governing equations

Abstract

State estimation is concerned with reconciling noisy observations of a physical system with the mathematical model believed to predict its behaviour for the purpose of inferring unmeasurable states and denoising measurable ones12. Traditional state-estimation techniques rely on strong assumptions about the form of uncertainty in mathematical models, typically that it manifests as an additive stochastic perturbation or is parametric in nature3. Here we present a reparametrization trick for stochastic variational inference with Markov Gaussian processesthat enables an approximate Bayesian approach for state estimation in which the equations governing howthe system evolves overtime are partially or completely unknown. In contrast to classical state-estimation techniques, our method learnsthe missingterms in the mathematical model and a state estimate simultaneously from an approximate Bayesian perspective. This development enablesthe application of state-estimation methodsto problems that have so far proved to be beyond reach. Finally, although we focus on state estimation, the advancements to stochastic variational inference made here are applicableto a broader class of problems in machine learning.