settings of (xt). These experiments are included in the process of tuning the parameters and training the functional form of the discrepancy bias. Therefore, the forecasting metric calculated at tested settings, E(xt), should be representative of the remaining uncertainty in the parameters and discrepancy bias. However, at settings other than xt, this calibrated and bias corrected model will not maintain the same level of fidelity, the lack of which would be captured by the proposed forecasting error term. Because simulation models as well as associated discrepancy bias are statistical processes, the forecasting error will be stochastic in nature. However, the forecasting error can be treated as a metric by aggregating uncertainties in various ways such as through the mean and standard deviation statistics, or the RMS norm. In a more conservative approach, the maximum forecasting error a simulation model yields while predicting multiple experiments can be used to determine the upper bound of forecasting metric. The way the forecasting errors are converted into a metric can be determined by the specific application, i.e. if the model is used to forecast a family of predictions with statistical relevance versus the model is exercised to predict at a single critical setting. In any treatment, as more experimental measurements become available to model calibration, the forecasting metric is expected to be reduced. In other words, provided the numerical model exhibits usefulness, either experimental campaigns can be concluded upon the quantification of predetermined acceptable forecasting error, or forecasting error can be quantified at the conclusion of strictly budgeted experimental campaigns. The following section discusses a case study which is a non-trivial, realistic application of the procedure. 5 DATA-RICH CASE STUDY: THE PRESTON-TONKS-WALLACE MODEL FOR TANTULUM In this section, we illustrate the quantification and control of the forecasting metric for a simulation model that is accompanied with a rich experimental campaign. The Preston-Tonks-Wallace (PTW) model of plastic deformation represents plastic stress in a material as a function of strain, temperature, and strain rate [4]. The PTW model requires seven dimensionless material dependent parameters to be defined, which must be inferred from the experiments for every metal. Even though the PTW model is applicable to make predictions over a wide range of control parameter triplets, no single setting of these seven model parameters can identically reproduce multiple experimental datasets distributed over the domain of applicability. The discrepancy arises perhaps due to some underlying assumptions of the PTW model. The PTW model for instance, assumes plastic stress is independent of the history of material loading. Moreover, the PTW model ignores non-isotropic plasticity as well as the material texture effects. 439
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