Calibration of multi-physics computational models using Bayesian networks

This paper develops a Bayesian network-based method for the calibration of multi-physics models, integrating various sources of uncertainty with information from computational models and experimental data. We adopt the Kennedy and OHagan (KOH) framework for model calibration under uncertainty, and develop extensions to multi-physics models and various scenarios of available data. Both aleatoric uncertainty (due to natural variability) and epistemic uncertainty (due to lack of information, including data uncertainty and model uncertainty) are accounted for in the calibration process. Challenging aspects of Bayesian calibration for multi-physics models are investigated, including: (1) calibration with different forms of experimental data (e.g., interval data and time series data), (2) determination of the identifiability of model parameters when the analytical expression of model is known or unknown, (3) calibration of multiple physics models sharing common parameters, which enables efficient use of data especially when the experimental resources are limited. A first-order Taylor series expansion-based method is proposed to determine which model parameters are identifiable. Following the KOH framework, a probabilistic discrepancy function is estimated and added to the prediction of the calibrated model, attempting to account for model uncertainty. This discrepancy function is modeled as a Gaussian process when sufficient data are available for multiple model input combinations, and is modeled as a random variable when the available data are limited. The overall approach is illustrated using two application examples related to microelectromechanical system (MEMS) devices: (1) calibration of a dielectric charging model with time-series data, and (2) calibration of two physics models (pull-in voltage and creep) using measurements of different physical quantities in different devices.

Quantitative model validation techniques: new insights

This paper develops new insights into quantitative methods for the validation of computational model prediction. Four types of methods are investigated, namely classical and Bayesian hypothesis testing, a reliability-based method, and an area metric-based method. Traditional Bayesian hypothesis testing is extended based on interval hypotheses on distribution parameters and equality hypotheses on probability distributions, in order to validate models with deterministic/stochastic output for given inputs. Two types of validation experiments are considered - fully characterized (all the model/experimental inputs are measured and reported as point values) and partially characterized (some of the model/experimental inputs are not measured or are reported as intervals). Bayesian hypothesis testing can minimize the risk in model selection by properly choosing the model acceptance threshold, and its results can be used in model averaging to avoid Type I/II errors. It is shown that Bayesian interval hypothesis testing, the reliability-based method, and the area metric-based method can account for the existence of directional bias, where the mean predictions of a numerical model may be consistently below or above the corresponding experimental observations. It is also found that under some specific conditions, the Bayes factor metric in Bayesian equality hypothesis testing and the reliability-based metric can both be mathematically related to the p-value metric in classical hypothesis testing. Numerical studies are conducted to apply the above validation methods to gas damping prediction for radio frequency (RF) microelectromechanical system (MEMS) switches. The model of interest is a general polynomial chaos (gPC) surrogate model constructed based on expensive runs of a physics-based simulation model, and validation data are collected from fully characterized experiments.