Empirically Improving Model Adequacy in Scientific Computing

Abstract

In developing mechanistic models, we establish assumptions regarding aspects of the system behavior that are not fully understood. Such assumptions in turn may lead to a simplified representation or omission of some underlying phenomena. Although necessary for feasibility, such simplifications introduce systematic bias in the model predictions. Often times model bias is non-uniform across the operational domain of the system of interest. This operational domain is defined by the control parameters, i.e., those that can be controlled by experimentalists during observations of the system behavior. The conventional approach for addressing model bias involves empirically inferring a functional representation of the discrepancy with respect to control parameters and accordingly bias-correcting model predictions. This conventional process can be considered as experimental data fitting informed by theoretical knowledge, only providing a one-way interaction between simulation and observation. The model calibration approach presented herein recognizes that assumptions established during model development may require omission or simplification of interactions among model input parameters. When prediction accuracy relies on the inclusion of these interactions, it becomes necessary to infer the functional relationships between the input parameters from experiments. As such, this study demonstrates a two-way interaction in which theoretical knowledge is in turn informed by experimental data fitting. We propose to empirically learn previously unknown parameter interactions through the training of functions emulating these relationships. Such interactions can be posed in the form of reliance of model input parameter values on control parameter settings or on other input parameters. If the nature of the interactions is known, appropriate parametric functions may be implemented. Otherwise, nonparametric emulator functions can be leveraged. In our study, we use nonparametric Gaussian Process models in the Bayesian paradigm to infer the interactions among input parameters from the experimental data. The proposed approach will equip model developers with a tool capable of identifying the underlying and mechanistically-relevant physical processes absent from engineering models. This approach has the potential to not only significantly reduce the systematic bias between model predictions and experimental observations, but also further engineers’ knowledge of the physics principles governing complex systems.