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Appl. Sci. 2020, 10, 4999 7 of 21 As shown in the sketch, the intermediate matrix can be obtained by input with padding around the original input matrix. Additionally, then the elements of output are the multiplication results of kernel and corresponding input elements moving at a specified stride. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 24 Figure 2. A simple sketch of the deconvolutional layer. 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. As shown in the sketch, the intermediate matrix can be obtained by input with padding around the original input matrix. Additionally, then the elements of output are the multiplication results of kernel and corresponding input elements moving at a specified stride. Figure 2. A simple sketch of the deconvolutional layer. Figure 2. A simple sketch of the deconvolutional layer. Convolutional n w re and more popular in computer vision [33,34], nature language [35], d 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. As shown in the sketch, the intermediate matrix can be obtained by input with padding around the original input matrix. Additionally, then the-Inepultemen-tKsernoelf outp-Outptutare the multiplication eur al n et ork sb eca me mo an corr esp ond ing inp ut le ent results of kernel and e m s moving at a specified stride. -Convolutional result FFi iggu ur ree 33. . A A ssi im mp pl le e ssk ke et tcch h o of f t th hee cco on nv vo ol lu ut ti io on naal l l laayyeer r. . With the complexity of application scenarios, the convolutional neural network goes deeper and With the complexity of application scenarios, the convolutional neural network goes deeper and deeper. However, some obstacles such as degradation of training accuracy and vanishing/exploding deeper. However, some obstacles such as degradation of training accuracy and vanishing/exploding gradients arise with deeper layers. In order to avoid the above problems, some outstanding means, data gradients arise with deeper layers. In order to avoid the above problems, some outstanding means, normalization, intermediate normalization layers [36], and Residual Neural Network (ResNet) [37], data normalization, intermediate normalization layers [36], and Residual Neural Network (ResNet) were applied in our approach. In this study, input and output were normalized to (−1, 1) by maximum [37], were applied in our approach. In this study, input and output were normalized to (−1, 1) by and minimum normalization, and batch normalization was applied after every deconvolution or maximum and minimum normalization, and batch normalization was applied after every -Input -Kernel -Output convolution except for the output layer. The detailed architecture of our two-stage algorithm is listed in deconvolution or convolution except for the output layer. The detailed architecture of our two-stage -Convolutional result Table 1, in which Deconv2d means deconvolutional operation, Conv2d means convolutional operation, algorithm is listed in Table 1, in which Deconv2d means deconvolutional operation, Conv2d means k is the size of kernel, s is the stride size, c is the channel size of output, in is the input size of linear layer, convolutional operation, k is the size of kernel, s is the stride size, c is the channel size of output, in Figure 3. A simple sketch of the convolutional layer. and out is the output size of linear layer. For more convenient description, the building block in ResNet is the input size of linear layer, and out is the output size of linear layer. For more convenient is separated to the basic block (a pair of 3 × 3 filters) and shortcut (connection operation with identify) descrWipitihonth, ethceombupiledxinitgy obfloacpkpilnicaRteiosnNsecteinsasreiposa,rtahtedcotnovtohleubtiaosniaclbnleouckra(lanpetawiroorkf 3go×e3s dfieltepres)r and in this study. As shown in Table 1, the input of the field reconstruction model is firstly reshaped to d s h e o e p r t e c r u . t H ( c o o w n e n v e e c r t i , o s n o mo p e e o r a b t s i t o a n c l we s i t s h u c i d h e a n s t i d f y e ) g i r n a d t h a i t s i o s n t u o d f y t . r a A i n s i s n h g o wa c n c u i n r a T c a y b a l n e d 1 , v t a h n e i i s n h p i n u g t / o e f x t p h l e o d f i i e n l d g a feature of large size by linear layer for subsequent deconvolutional operations. The size of output grercaodniesntrtuscatrioisnemwiotdhedleiespfeirsltalyeress.hInapoerdetrotoaafevaotiudrethoefalbaorvgepsirzoeblbeymsli,nseoamrelaoyuetrstfaonrdsiunbgsmeqeuaennst, features become a specified 256 × 64 × 4 after the transformation of six deconvolutional layers and daectaonvoormluatiloiznatlioonp,eirnatteiromnse.dTiahte nsiozremoafliozuatipountlfaeyaetrusre[3s6b],eacnomd eReasisdpueaclifNieedu2ra5l6N×e6tw4 o×rk4 (aRfteesrNtehte) then the output features are interpolated to 256 × 64 × 4 and 256 × 64 × 4 for physical fields of stator [t3ra7n],sfwoermreaatipopnlioefdsixndoeucronavpoplruotaiochn.alInlatyheirsssatnudyth, einpthuet oanudtpouut tfpeauttuwreesraerenoinrmterapliozleadtetdoto(−215, 61)×b6y4 and rotor blades. The similar interpolation operation can be found in performance prediction model. m× 4axainmdu2m56 ×an6d4 ×m4ifnoirmpuhmysicnaolrfmiealdlizsaotfiosnta,toarnadndbraotctohr bnloardmesa.liTzhaetiosinmiwlaarsintaeprploieladtioanftoeprereavteioryn The physical fields with different sizes are adjusted to the specified size of 256 × 64 by an interpolation dcaencobnevfoluntidoninorpecorfnovrmoluanticoenperxecdeipctifoonr mthoedoeul.tpTuhtelapyheyrs.iTcahlefideeldtasilwedithardchififtercetnutresiozfesouaretwadoj-ustsategde operation which makes the performance prediction model away from the affection of input size. atolgtohreitshpmecifsielidstseidzeinofT2a5b6le× 614, ibnywanhicnhteDrpeocolantvio2ndomperaantsiodnewcohnivcholmutaioknesalthoeppereartfiornm, aCnocnevp2rdedmicetaionns After the subsequent convolutional operations from layer 1 to layer 6, the average pooling [38] and cmoondveolluatwioanyalfropmerathtieona,ffkecitsiothneosfizienpouf tkesirzne.l,Asftiserththeestsruidbesesqizuee,ncticsotnhveoclhuatinoneallsoizpeeoraftoiountps ufrto, imn linear layers are adopted to obtain objective output from extracted features. In addition, the active ilsaytehre1intopulatyseirze6, othf elinaveaeralgaeyepro, oalnindg o[3u8t] iasntdhelinoeuatrpulatyseirzseaoref alidnoeparteldaytoero. bFtoarinmoobrjectciovnevoeuntipeunt function ReLU [39] is employed to enhance the nonlinear performance of the deep convolutional description, the building block in ResNet is separated to the basic block (a pair of 3 × 3 filters) and neural networks. shortcut (connection operation with identify) in this study. As shown in Table 1, the input of the field reconstruction model is firstly reshaped to a feature of large size by linear layer for subsequent deconvolutional operations. The size of output features become a specified 256 × 64 × 4 after the transformation of six deconvolutional layers and then the output features are interpolated to 256 × 64 × 4 and 256 × 64 × 4 for physical fields of stator and rotor blades. The similar interpolation operation can be found in performance prediction model. The physical fields with different sizes are adjusted to the specified size of 256 × 64 by an interpolation operation which makes the performance prediction model away from the affection of input size. After the subsequent convolutional operations fromPDF Image | Performance Prediction of a S-CO2 Turbine
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