Uncertainty quantification and sensitivity analysis of Parameterization-free surrogate model

Surrogate model based optimization method is widely-used to accelerate the design and optimization process [1]. The input of regression model used in the surrogate model are numbers, which requires users to parametrize the geometries. In this paper, a new parameterization-free surrogate model is introduced and its corresponding uncertainty quantification and sensitivity analysis method are discussed. The input of new surrogate model methods is surface mesh of simulation domain. Graph Neural Networks (GNNs) is used to extract geometric information, and Convolutional Neural Networks (CNNs) is used to predict contours. This framework bypasses parameterization,as a consequence, uncertainties introduced by manual parameterization is reduced. However, such changes compared with conventional surrogate model methods impose great challenge on uncertainties quantification and sensitivity analysis. Uncertainties quantification in this paper means the error bar of prediction results, which is calculated by Gaussian Process Regression method in current surrogate method. In this paper, a new quantification method achieved by Kullback-Leibler divergence (KLD) is introduced. And the sensitivity analysis is conducted by Automatic Differentiation, which gives a Jacobian matrix of inputs. The method and analEmail addresses: jc980@cam.ac.uk (Jiajun Cao∗), ql295@cam.ac.uk (Qingbiao Li∗), lpx1@cam.ac.uk (Liping Xu), yangrui@shanghai-electric.com (Rui Yang), daiyj3@shanghai-electric.com (Yuejin Dai) Preprint submitted to Journal of LTEX Templates January 22, 2022 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 January 2022 doi:10.20944/preprints202104.0762.v3 © 2022 by the author(s). Distributed under a Creative Commons CC BY license. ysis mentioned above are demonstrated by a low-pressure steam turbine rotator and its exhaust system.


Introduction
Despite the rapid increase of renewable energy production, around 60% of the world electricity will still be generated by fossil fuel fired power plants by 2040 [2]. (solar,wind and tidal power). As a consequence, maintaining efficiency when operational condition varies have become a big concern in the design of some steam turbines. To further improve the operational flexibility and efficiency of steam tur-10 bine, exhaust hood loss is an important part that cannot be neglected. As Fig 2 shows, LP exhaust loss is the second highest loss among all losses. And despite the LP blade loss is still the highest, the actual margin for further improvement is rapidly diminishing after intensive research and development efforts in the past decades. In contrast, exhaust hood has relatively simpler structure. Its 15 aerothermal performance can be improved with less concerns from mechanical aspects. On the other hand, the multi-objective optimization of low-pressure steam turbine exhaust system is quite expensive due to the large size of simulation domain and multiple workload conditions. And the design target could be frequently changed according to the user specifications, which makes surrogate 20 model more beneficial to reduce the computational cost by utilizing numerical simulation results of previous optimizations.
In the recent decades, a lot of surrogate model methods appeared, which can be loosely categorized into: (1) Polynomial Response Surface Method [5]; (2) Kriging Model [6]; (3) Radial Basis Function and Extended Radial Basis 25 Function [7]; (4) Artificial Neural Network [8]; (5) Support Vector Machine [9]. Fig ?? shows the process of existing surrogate model method, and one key step in this process is manual parameterization, which is to choose some important geometric parameters according to the experience of researchers to describe the geometry . In this step, if too few parameters were used, some geometric 30 information will be lost because it is insufficient to describe geometric details with high-order surfaces (e.g., airfoils and blades). On the other hand, using too many parameters or choosing irrelevant parameters will cause problems of overfitting [10]. It is recognized by the authors that manual parameterization is the bottleneck that prevents further improvement of the performance of surrogate 35 model method in terms of both accuracy and flexibility.
Recently, some neural networks and network structures have been applied to the field of computational mechanics, which inspired the idea of parameterfree surrogate model method. A widely used neural network is the Physical Information Neural Network (PINN). It has been applied to solid mechanics [11] 40 and fluid mechanics [12]. PINN is built with multilayer perceptrons. Its inputs are still manually defined parameters. Another widely used neural network, Convolutional Neural Network (CNN), can predict two-dimensional data [13], but cannot extract geometric information. As a popular network structure, autoencoders can compress and reconstruct data [14]. Combined with CNNs, 45 autoencoders can extract information from images and reconstruct images. This paper proposes a new surrogate model method that uses GNNs, and a Conditional Variational Autoencoder (CVAE). Figure ?? shows the differences between numerical simulations used as analysis tools, existing surrogate model methods, and the proposed new surrogate model method. The new method 50 can process the boundary surfaces of fluid domains from surface meshes used in numerical simulations and extract relevant geometric features according to their importance to the results. Compared to existing surrogate models, the new method contains less uncertainty introduced by manual parameterization. This new approach also allows the use of different types of designs from different 55 sources, as the geometric input to the model is a surface mesh, rather than user-defined parameters. In addition, the new method is able to predict the twodimensional distribution of variables (in the form of contour plots) by processing the image with CNNs.
The ability to predict a two-dimensional distribution of variables is achieved 60 by applying CNNs. Through the combination of convolutional layers, information can be extracted from the image and the convolutional results can be identified using a multilayer perceptron [15]. In this study, CNNs is used to predict contours from the underlying distribution.
The ability to use face meshes is achieved by applying GNNs. The graph operation nature of GNNs enables it to handle non-Euclidean domains by defining the connectivity of grid points, whereas CNNs can only handle regular Euclidean data like graphs. In existing GNN variants [16], GNNs are divided into three types: cyclic GNNs, spatial GNNs and spectral GNNs. In this study, the surrogate model is built based on Spectral GNNs because of its advantages in GNNs is based on signal processing theory. The key step, the convolution operation, is done by the Chebyshev polynomial approximation [18]. In this study, GNNs is used to extract geometric information by optimizing parameters in a neural network through backpropagation of losses. This is to select relevant 75 information based on the feedback of prediction error, avoiding geometric information loss and overfitting problems. Furthermore, since GNNs are able to handle non-Euclidean data, the input to the new surrogate model can be both unstructured grids and structured grids.
The framework for this new surrogate model approach is CVAE. It is an ex-80 tension of Variational Autoencoder (VAE). VAE is an extension of autoencoders.
Its latent distribution is regularized to increase the power of interpolation and extrapolation [19]. On this basis, conditions are added to the underlying distribution to divide the samples into different groups. In this study, CVAE was used to normalize and classify the design space.

Methodology
The present surrogate model consists of two parts: optimization part (Sec. 2.1) and machine learning part (Sec. 2.2). The optimization part is based on Genetic Algorithm (GA), which is to optimize the performance of LPES at five workload conditions. At the same time, the results of optimization are used as the 90 database for the surrogate model because the designs generated by the optimizer are similar to designs that will appear in future optimization. In this study, the optimization part is only introduced as a tool for accumulating database.

Optimization
For optimization, the surrogate model is used to evaluate the performance of 95 designs, which will replace numerical simulation and predict the performance of the design. But when the design is out of existing design space of the database, this optimization iteration still needs the results of additional numerical simulations and then the new results will be added to the surrogate model. Therefore, ideally, the optimization will run in the hybrid mode shown in

Machine Learning
The machine learning module is used for training the surrogate model. It consists of two main parts: mesh encoder and conditional variational contour decoder.

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The most important layers used in the mesh encoder is the fast spectral convolution layer. The mesh convolution operator * is defined as a Hadamard product in Fourier space: To reduce the computational cost, convolution is conducted by a kernel g θ with Chebyshev polynomial of order K.
And θ k are the coefficients of the Chebyshev polynomials. T k can be expressed as: with the initial condition T 0 = 1 and T 1 = x. This represents a Chebyshev polynomial of order K.
With the mesh filter shown above, the fast spectral convolution layer can be expressed as the following equation with n×F in input and n×F out output where y j means the j th feature.
Another important layer used in the mesh encoder is the mesh sampling layer, which includes the down-sampling layer and up-sampling layer in autoen-110 coder [20]. In the encoder, important information is chosen by down-sampling layer and irrelevant information will be discarded. The convolution layers used in this study represents mesh in multi-scales so that mesh sampling layer can capture the local and global geometric information. The down-sampling operation is conducted by a transform matrix Q down ∈ {0, 1} n×m , where m is the number of vertices in the original mesh and n is the number of vertices in the down-sampled mesh. Q down (p, q) = 1 means the q-th vertex is kept during the down-sampling, while Q down (p, q) = 0 means the vertex is discarded. The transformation matrix is optimized to minimize the surface error by quadric matrices [21].

Contour decoder is built by CNNs-based CVAE. AutoEncoder (AE) uses
CNNs to compress graphical data to a latent vector and then reconstruct the graph with the latent vector. The neural network is trained to reconstruct the graphs with less error. Variational AutoEncoder (VAE) uses variational inference to estimate the latent vector rather than directly encoding from input graph [22]. The latent vector z can be estimated by observation vector x using the following equation: However, p(z|x) is usually very difficult to compute directly [19]. Therefore, another distribution is used to approximated p(z|x) in the training process.
The Kullback-Leibler divergence is used to measure the difference between two probability distributions, which is to be minimized during the training process.

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CVAE adds conditions into the latent distribution so that different classes of input data are categorized into different groups. In this study, conditions (blade passages index) are added into the latent distribution twice to label the input data.
In the CVAE, the contour decoder used Residual neural Network (ResNet), 130 a type of classical artificial neural network, which is inspired by pyramidal cells in the cerebral cortex. ResNet simulates this structure by building shortcuts to skip some layer, rather than passing information layer by layer. Fig 4 shows a basic block of ResNet. F (x) is to fit the residual between x and target mapping H(x) rather than directly fitting H(x). It is easier for the optimizer to minimize 135 the residual to zero [23]. In this study, more hidden layers are added to fit the highly non-linear relationship between input and output, but the performance of neural network decreases rapidly with more layers. To solve the degradation problem, ResNet is adapted because it can pass information from front layers to rear layers, which reduces the loss of information in the hidden layers.

Contour Decoder Blocks
Contour decoder consists of two types of blocks: ResNet block and basic MSE measures the average of pixel-wise error between the contours predicted by model (Y i ) and contours predicted by numerical simulation ( Y i ). It is defined mathematically by: KLD [24] measures the difference between one probability distribution and the reference probability distribution. In variational autoencoder, KL loss is the sum of all the KLD between the components in latent distribution and the standard normal distribution. With minimizing the KL loss, the latent distribution is closer to the standard normal, which can improve the interpolation and extrapolation ability of the surrogate model. KLD can be defined by: As it is to measure the KLD between the components in latent distribution and the standard normal (σ 2 = 1, µ 2 = 0), it can be simplified as the following equation for convenience: where µ is the mean vector, σ is the variance vector.
Structural similarity loss, or Structural Similarity Index Measure (SSIM) [25], is a method to measure the similarity between two figures. Here, it is used to optimize the neural network to make predicted contours and simulated contours more structurally similar. It is defined by: where µ Yi , µ Yi are the mean of Y i and Y i , σ 2 Yi , σ 2 Yi are the variances of Y i and Y i , σ Yi Yi is the covariance of Y i and Y i , c 1 , c 2 are two variables to stabilize the division with weak denominator.
Finally, the loss function of the surrogate model is defined by the following equation: where three coefficients k 1 , k 2 and k 3 are user-defined hyperparameters.

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The researches about LP steam turbine exhaust system began in the late 1960s. In the early stage, research community has performed 1D analysis by theoretical analysis or experiments, which has proved that conventional diffuser data is not applicable to LP exhaust system.
In the recent decades, 3D numerical simulation has been applied in studying the flow features in the LP exhaust system, which gives a deeper understanding about the factors that influence the performance of system. The literature review written by Burton [? ] has summarised the achievements by new research tools in the recent years.
Since this report is about the inlet flow condition of LP exhaust system at 230 part load condition, the literature review, therefore, will more focus on interaction between last stage and hood and part load performance.

Flow features in the LP exhaust system
Identifying flow features is the first step to understand the physics and loss mechanism in the LP exhaust system because flow features like separations and 235 shock waves will largely affect the performance of the system.There are many literature about the flow features inside the exhaust system and the reason why these flow features form, which help researchers to further understand the source of loss.
The flow inside the LP exhaust system is three-dimensional and complicated.

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But generally, it can be divided into two separate parts: axial-to-radial flow in the diffuser and down flow to the condenser At the inlet of diffuser, the flow is highly non-uniform in radial direction and circumferential direction, which is generated by last stage and affected by the axis-asymmetric flow. In addition, the high turning angle of diffuser also 245 contributes to the separations in the LP exhaust system. Fig ?? shows the flow features of axial-to-radial flow, including bearing cone separation, flow guide separation, flow guide tip separation and additional vortices.

Simulation Setup
The solver used for the numerical simulations is Ansys CFX, which is a 250 widely-used commercial CFD solver for the research community and industry.
The simulation is a Reynolds-averaged Navier-Stokes (RANS) simulation, which uses k − ϵ turbulence model [27].  The inlet boundary condition is applied at the inlet of the simulation domain, which has total pressure and total enthalpy. The flow direction at the inlet is 255 normal to boundary. And the outlet of the simulation domain is at the end of extension, which is applied with static pressure boundary conditions. The flow direction at the outlet is also normal to boundary. To perform the part-load simulations, the total pressure is reduced at the inlet to reduce the mass flow rate and work output. The static pressure at the outlet is 6.2kPa due to the  Each of them generates inlet boundary condition for a 90-degree section of exhaust hood.

Processing of Numerical Simulation Results
The objective of GA-based optimization is to increase power output of the 275 last two low pressure steam turbine stages. It is calculated by the difference between the total energy fluxes pass through inlet and outlet of the last two stages, which is summed value of all the elements of contour map of energy flux.
Assuming the system is adiabatic, the power of each element is obtained by the product of local total enthalpy and local mass flow rate: Since the inlet boundary condition is known in the simulation, only the total enthalpy contours and mass flow rate contours at the outlet of the last two stages are extracted from numerical simulations to generate power contours as shown in Equation 12, which will also be the output of the surrogate model. Because of the multiple mixing plane method, there are four blade passages for each 285 simulation, and five workload conditions for each design. Admittedly, there is certainly some uncertainties in the numerical simulations, but it is not primary concern in this paper since the key of this study is to develop a new surrogate model method.

Test Setup
To test the surrogate model, 32 cases are randomly selected from 582 cases.
Therefore, there are 640 contours to predict in total because each case has 4 blade passages and 5 workload conditions. And during the training, there is a k-fold cross validation, which means 55 cases of the remaining 550 cases will be 295 used for validation every 10 epochs. is also evaluated by three different metrics: • Mean squared error is used to measure the average of pixel-wise error between the model prediction contour (Y i ) and the ground truth contour • SSIM is used to evaluate the structure similarity level between the model 310 prediction contour (Y i ) and the ground truth contour ( Y i ). SSIM is an index between 0 and 1, where the SSIM = 1 two contours are identical; • Summed value error (SVE) measures the differences of summed value, which is defined by the following equation: where y i is the value of ith pixel of predicted contours, and y i is the value of ith pixel of target contours. This error also indicates the error in the predicting averaged value, like averaged pressure, averaged temperature, 315 averaged velocity of a surface.

Discussion
The new surrogate model method established a mapping relationship between the surface mesh of fluid domain and two-dimensional distribution of 320 flow variables. The application of this new method can be extended beyond the area of aerodynamics optimization. Since it can process both structural and unstructural mesh, it is also applicable in various problems in different fields, which are solutions of partial differential equations, traditionally using finite element analysis and electromagnetic analysis methods.

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This method can also be used as an inverse method. It can be achieved by exchanging the input and output of the mapping relationship built in this paper.
To be more specific, users can import the desired two-dimensional distributions of physical properties into the contour encoder, and the designs could be created by the mesh decoder. This approach also exhibits a high degree of flexibility and compatibility.
Since the input to the new method is a face mesh, it can adopt the same topology as the database geometry. This means that it is compatible with geometries defined by different parametric methods. This is useful for the ability to further 340 increase the size of the proxy model database using variable data sources.
Compared to existing surrogate model methods, this new method can also predict the two-dimensional distribution of variables (contours) based on face meshes. Outlines can help designers identify physical mechanisms, improve designs, and serve many other purposes. In testing, the average similarity score