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Appl. Sci. 2020, 10, 4999 9 of 21 αt ←α· θt ← θt−1 − αt·mˆ t/( vˆt + ε) 1−βt2(1−βt1) (20) (21) where subscript t indicates the iteration step of the optimization process, l is the loss function which can be lstage1 or lstage2, θ is the learnable parameters of the neural network (θ ∈ Θ1 or θ ∈ Θ2), m is the first moment estimation, v is the second moment estimation, β1 and β2 are attenuation coefficients, α is the learning rate, and ε is a small number which prevents division by zero in the implementation. In this study, ε = 10−8, β1 = 0.9, and β2 = 0.999. The optimization in stage 1 is performed such that the defined loss is minimized. Firstly, the field loss in this study is the mean square error (MSE) between the predicated field fˆ and the original field f, which can be written as: C l l 1 N C l l lf = E{x,f}∼Γ( ||fˆi − fi||) ≈ N ||fˆi − fi|| (22) l=1 i=0 l=1 where N is the sample size and C is the number of fields in the train dataset. The subscript i represents sample index and the superscript l is the type of the field. To circumvent a very blurred predication only by the field loss, the absolute error of gradient information between the predicated field and ground truth is reckon in loss function, also called gradient loss, that is defined as: C l l 1NC l l l∇ = E{x,f}∼Γ( ||∇·fˆi − ∇·fi||) ≈ N ∥ ∇·fˆi − ∇·fi ∥ (23) l=1 i=1 l=1 Then the total loss function to be minimized can be written as a combination of field loss and gradient loss, where λ is the loss weight. In this study, λ = 0.1. lstage1 = lF + λl∇ (24) The optimization in stage 2 was performed to minimize MSE. For the batch of N samples, MSE of the parameterization case can be defined as: lstage2 = N (yti − yi ) (25) 1N p2 m=0 R square value (R2), mean absolute error (MAE), and root mean squared error (RMSE) are adopted to compare efficiency prediction results. R2, MAE, and RMSE are calculated using Equations (26)–(28), where Prei and Acti represent the predicted and actual efficiency and N is the number of observations in the testing dataset. NN2 R2 (Pre, Act) = (Prei − Acti )2 /Acti − Act (26) i=1 i=1 1 N MAE(Pre, Act) = N 1 N RMSE(Pre, Act) = N i=1 |Prei − Acti| (27) i=1 (Prei − Acti )2 (28)PDF Image | Performance Prediction of a S-CO2 Turbine
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