  This paper introduces a novel hierarchical Bayesian model specifically
designed to address challenges in Inverse Uncertainty Quantification (IUQ) for
time-dependent problems in nuclear Thermal Hydraulics (TH) systems. The unique
characteristics of time-dependent data, such as high dimensionality and
correlation in model outputs requires special attention in the IUQ process. By
integrating Gaussian Processes (GP) with Principal Component Analysis (PCA), we
efficiently construct surrogate models that effectively handle the complexity
of dynamic TH systems. Additionally, we incorporate Neural Network (NN) models
for time series regression, enhancing the computational accuracy and
facilitating derivative calculations for efficient posterior sampling using the
Hamiltonian Monte Carlo Method - No U-Turn Sampler (NUTS).
  We demonstrate the effectiveness of this hierarchical Bayesian approach using
the transient experiments in the PSBT benchmark. Our results show improved
estimates of Physical Model Parameters' posterior distributions and a reduced
tendency for over-fitting, compared to conventional single-level Bayesian
models. This approach offers a promising framework for extending IUQ to more
complex, time-dependent problems.
