In many areas of engineering and physics, accurately predicting the behavior of complex systems requires numerical models, often described by nonlinear differential equations or phenomenological representations. These models depend on parameters that must be carefully calibrated to ensure reliable predictions. Calibration becomes particularly challenging when the system is dynamic, non-stationary, and nonlinear, and when the control actions characterizing its behavior are only partially known. For example, we can think of a train operating under not perfectly known driver commands or a building thermally regulated by a thermostat system that is not fully documented. In such contexts, Bayesian calibration provides a promising framework to infer unknown parameters while accounting for uncertainties in both measurements and environmental conditions. This framework allows the integration of prior knowledge and quantification of uncertainty in the posterior distributions of the calibration parameters. However, applying Bayesian calibration to non-linear dynamic systems with partially known inputs is difficult: likelihood evaluations often require expensive numerical simulations, and the posterior must integrate over the high-dimensional, partially observed control variables, leading to intractable distributions. The objective of this work is to address these challenges by coupling statistical learning and computational statistics. Specifically, we use Gaussian Process Regression (GPR) to construct surrogate models of the expensive log-likelihood function, capturing both its value and gradient. This surrogate enables efficient exploration of the parameter space while accounting for uncertainty in model outputs. We further combine GPR with the Metropolis-Adjusted Langevin Algorithm (MALA), a gradient-informed Markov Chain Monte Carlo method well suited to high-dimensional and complex posterior distributions. Together, this approach allows efficient and robust Bayesian inference of calibration parameters in dynamic, nonlinear, and partially observed systems. We demonstrate the potential of this methodology first on an analytical test case where all model parameters are controlled, and then on two industrial applications, a first one in railway dynamics and a second one in thermal evaluation of buildings.