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Appl. Sci. 2020, 10, 4999 6 of 21 In stage 1, the deconvolutional neural network is employed to establish the reconstruction mapping from design variables. The input is design parameters x, while the temperature and pressure fields are target physical fields. Assuming the reconstruction mapping in stage 1 can be defined as followed: fˆ = Fˆ1(x;Θ1) (10) where x is the input design variables, fˆ is the reconstructed field, Θ1 is the learnable parameters in reconstruction deconvolutional neural network, the reconstruction mapping Fˆ 1 can be obtained by minimizing the expectation of loss function lstage1 in the definition domain of the dataset. The training process of stage 1 is presented as: Θ1 = arg min E{x,f}∼D lstage1 (11) Θ1 where {x, f}∼D indicates design variables and fields samples obtained by numerical simulations in definition domain D. At stage 2, the performance is predicted from physical fields using a deep convolutional neural network. In this study, the input at stage 2 is the reconstructed fields obtained at stage 1 and the output is the interested performance of S-CO2 turbine, power, and efficiency. The mapping function from physical fields too performance can be described as follows: ˆ ˆˆ ˆˆ ψ = F2 f;Θ2 = F2 F1(x;Θ1);Θ2 (12) where ψˆ is the predicted turbine performance, fˆ is the reconstruction field at stage 1, Θ2 is the learnable parameters of the deep convolutional neural network, the mapping function Fˆ 2 can be obtained by minimizing the expectation of loss function lstage2 in the definition domain of the dataset. The training process of stage 2 is formalized as: Θ2 = arg min E{f,ψ}∼D lstage2 (13) Θ2 where {f, ψ}∼D indicates fields and performance samples obtained by numerical simulations in definition domain D. It is obvious that the input design variables of stage 1 are low dimensional data while the physical fields with high dimension are obtained as output. Thus, deconvolutional neural network is utilized to expand low-dimensional input to high-dimensional fields. Deconvolutional neural network was first proposed by Zelier [28] and the general application was presented in their following works [29,30]. With the development of deconvolutional neural network, plenty of applications are conducted on scene segmentation [31], image processing [32], and so on. The deconvolutional operation is illustrated in Figure 2 with a simple example with padding size b = 1, stride size s = 2, and kernel size k = 3. For a more convenient description, the input, kernel, and output are marked in blue, gray, and green, respectively. The input of size 3 × 3 is interpolated withzeroandthesizeofintermediatematrixupto(s×3+b)×(s×3+b),thatis7×7. Thefinal output is the result of convolutional operation between the kernel and intermediate matrix with stride of 1. In this point, deconvolution can be seen as a kind of special convolution. Convolutional neural networks became more and more popular in computer vision [33,34], nature language [35], and so on due to their powerful ability of feature extracting and learning. It is a natural idea to utilize convolutional neural networks to extract the low-dimensional performance from the high-dimensional physical fields. In mathematics, the convolution operation is a kind of multiplication of input and kernel at certain strides, as shown in Figure 3. Similar to Figure 2, the input, kernel, and output in this convolutional example are marked in blue, gray, and green, respectively.PDF Image | Performance Prediction of a S-CO2 Turbine
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