Uncertainty Quantification for parameter estimation of an industrial electric motor using hierarchical Bayesian inversion

In this work, we employ hierarchical Bayesian inference to estimate aleatory parameter uncertainty of a “black-box” simulation model of an industrial electric motor using noisy measurements obtained from a real test bench. Standard sampling-based approaches, like Metropolis–Hastings (MH), require a huge amount of expensive simulation model evaluations in order to compute the likelihood. This is prohibitive from a computational point of view. Instead, Hamiltonian Monte Carlo (HMC) can be used to reduce the number of samples significantly. One key ingredient in HMC, however, is the availability of a function evaluating the gradient of the simulation model with respect to its parameters. In our problem setting, the simulation model is a “black-box” model — a situation which is common in many industrial engineering problems. There is no function available for evaluating the gradients, which therefore must be either approximated numerically or evaluated from a suitable analytical approximation of the simulation model using a surrogate. In this work, we introduce a new approach to enable HMC for complex black box simulations based on B-splines surrogates trained on spatially adaptive sparse grids. The B-spline surrogates are able to accurately represent the simulation model and in addition its gradients. We show how they can be used within HMC in order to infer the probability distribution of all five electric motor parameters accurately and efficiently, and we demonstrate its superiority in convergence compared to MH. •12 input parameters of a real motor test-bench are inferred from noisy measurements.•A hierarchical Bayesian formulation leads to uncertainty estimates for all results.•B-spline surrogates provide accurate gradients with few training data.•Using B-splines on sparse grids enables the usage of the No-U-Turn-Sampler (NUTS).•NUTS is significantly more efficient than the classical Metropolis–Hastings algorithm.