Current surrogate model methods that are widely used in optimization and design processes rely on manual parameterization to describe the geometry of objects. The loss of geometric information in this process limits the prediction accuracy of surrogate model. To tackle this problem, the new method directly picks important geometric features from surface meshes of fluid domain using Graph Neural Networks (GNNs) and predicts contours of fluid variables based on extracted information with Convolutional Neural Networks (CNNs). The prediction error of CNNs propagates backwards to train GNNs to select sensitive features from surface meshes. This framework reduces uncertainties introduced by manual parameterization and the loss of geometric information because the input of this new method is from the meshes used in the numerical simulations. With CNN and larger amount of extracted geometric information, this method can also predict higher dimensions distributions of flow variables rather than only several performance metrics. The nature of non-parametric representation of geometry also allows users to access designs defined by other parameterization methods to create a larger database. Additionally, thanks to the generic nature of the new method, it can be used for any other design or optimization processes governed by partial differential equations involving complicated geometries. To demonstrate this new method, a non-parametric surrogate model is built for a low-pressure steam turbine exhaust system (LPES). The new surrogate model uses 10 surfaces meshes of the LPES as input and it is used to predicts the energy flux contours at the exit of the last stage of the turbine. Altogether 582 designs have been generated, which contains two types of geometries defined by different methods. Among them, 550 cases are used for training, and 32 cases for testing. The power output of the last two stages of the turbine predicted by the surrogate model has average 0.86% difference compared with those of numerical simulations over a wide range of power ratings. The structural similarity index measure (SSIM) is used to measure the differences between the simulated and predicted contours at the exit of the last rotor, where the average SSIM of 640 contours is 0.9594 (1.0 being identical).

Surrogate model based optimization method is widely-used to accelerate the design and optimization process~\cite{marler2004survey}. 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. \Gls{gnns} is used to extract geometric information, and \Gls{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 analysis mentioned above are demonstrated by a low-pressure steam turbine rotator and its exhaust system.

No abstract