Clearly, Eq. (7) has redundant input variables since, according to Eq. (6), *σ* can be determined solely via the original input sets **x** = [*x*, $\phi $, *P*, *γ*, *H*] and **θ**. Based on the well-known tight physical relationship between *displacement* and *stress* fields, it is reasonable to conjecture that *σ* can also be accurately predicted as a function of the neighboring displacements (perhaps augmented with some subset of the inputs and parameters) with the important input *P* omitted. If we can identify such a predictive relationship empirically, this gives us a mechanism to use the bias-corrected displacement model to also correct for the bias in the stress model. To identify an appropriate predictive relationship and also an appropriate subset of input variables to augment the displacement inputs, we first generated a Latin Hypercube design with 2000 samples for all input variables in **x**, and then we conducted simulations of *w*, *w*_{1}, *w*_{2}, *w*_{3}, *w*_{4} and *σ* from the supplied tank model at each input combination. With these data serving as the training data, we then fit neural network regression models for predicting *σ* as a function of {*w*, *w*_{1}, *w*_{2}, *w*_{3}, *w*_{4}} and various subsets of the other input variables. The best model (i.e., with the highest predictive power) was of the form
Display Formula

(8)$\sigma =f(x,\phi ,H,w,w1,w2,w3,w4;\theta *)+\epsilon \sigma $

where **θ**^{*} denotes the MAP values of the model parameters obtained from model calibration and *ε*_{σ}_{}_{} ∼ *N* (0, MSE) is the neural network regression model error. To determine the best model, we set aside a test set of 600 simulation data points, separate from the training set (1400 simulation data points), and used the fitted models to predict the test set. The validation results for predicting the test set are shown in Fig. 14. The coefficient of determination between the predicted and actual *σ* for the test cases was **R** = 0.9969, which is a nearly perfect fit. The root mean square error was 3.0809 × 10^{2}, which indicates the amount of uncertainty by using the fitted neural network model for predicting *σ*. The learned relationship between *w* and *σ* in Eq. (8), together with the bias-corrected predictive model for *w*, will be used as a surrogate for the bias-corrected model for stress to generate *σ* predictions.