Bayesian calibration with missing data and model discrepancies: Application to high-speed train with uncertain control

• We calibrate dynamic models with uncertain and missing input data. • Our method models the log-likelihood using Gaussian Process regression. • The approach accounts for both model error and input uncertainty. • We reduce solver calls using a surrogate-based Bayesian framework. • A railway case study demonstrates robustness and efficiency. With the increasing complexity of engineering systems, the calibration of physical models has become a key step in ensuring predictive accuracy for industrial applications. In practice, however, calibration is often hindered by latent environmental variables whose influence on system behavior cannot be directly observed or controlled. Standard calibration techniques typically presume full knowledge of input variables — an assumption that, when violated, can lead to biased estimates and degraded model performance. This issue is exacerbated in dynamic systems, where both inputs and outputs are time-dependent functions, and measurements are inherently noisy and approximate. To address these limitations, we formulate the calibration problem within a Bayesian framework, treating the unknown environmental variables as random inputs with an associated probability distribution. Rather than optimizing calibration parameters for a fixed set of inputs, we compute their posterior distribution and marginalize over the distribution of the unobserved variables. This approach propagates input uncertainty through the model and provides a natural mechanism for regularizing model error, thereby enhancing both robustness and interpretability. We demonstrate the practical benefits of this method on a railway dynamics case study, where it outperforms standard calibration in terms of both accuracy and reliability under uncertainty.