This paper introduces a novel surrogate model for two-dimensional adaptive steady-state thermal convection fields based on deep learning technology. The proposed model aims to overcome limitations in traditional frameworks caused by network types, such as the requirement for extensive training data, accuracy loss due to pixelated preprocessing of original data, and inability to predict information near the boundaries with precision. We propose a new framework that consists primarily of a physical-informed neural network (PINN) and a graph convolutional neural network (GCN). The GCN serves as the prediction module and predicts thermal convection in the two-dimensional computational domain by considering the mutual influence between unstructured nodes and their neighbors. On the other hand, the PINN acts as the physical constraint module of the framework by embedding the control equation of thermal convection into the loss function of the neural network, ensuring that the inference and prediction results of the GCN comply with the constraints of the control equation. The advantages of this framework lie in two aspects. First, the computation mechanism of the GCN is more in line with the actual evolution of temperature fields. Second, the PINN enhances the cognitive ability of the surrogate model toward the convection field information. It accurately describes the changes of temperature gradient information at the boundary position and reduces the model's demand for training data. To validate the advantages of the proposed model, we gradually analyzed the model's geometric adaptability and predictive accuracy from the single cylinder case to the double cylinder case. We also investigated the impact of the number of sampling points on model training and compared the model's prediction results with those of a purely data-driven model. The results show that the proposed model exhibits good geometric adaptability and stability. With only 20 training data, the mean error of the proposed model in predicting the velocity and temperature field is less than 1% and 0.6% for the single cylinder, and less than 2% and 1% for the double cylinder case, while the mean error of the purely data-driven GCN model in predicting the velocity and temperature field is 9.4% and 6.4% for the double cylinder case. These findings demonstrate the effectiveness of the proposed physics-informed graph convolutional neural network, allowing for more accurate prediction of fluid flow and heat convection using surrogate model.

We develop a deep neural network-based reduced-order model (ROM) for rapid prediction of the steady-state velocity field with arbitrary geometry and various boundary conditions. The input matrix of the network is composed of the nearest wall signed distance function (NWSDF), which contains more physical information than the signed distance function (SDF) and binary map; the boundary conditions are represented by specifically designed values and fused with NWSDF. The network architecture comprises convolutional and transpose-convolutional layers, and convolutional layers are employed to encode and extract the physical information from NWSDF. The highly encoded information is decoded by transpose-convolutional layers to estimate the velocity fields. Furthermore, we introduce a pooling layer to innovatively emphasize/preserve information of boundary conditions, which are gradually flooded by other features during the convolutional operation. The network model is trained using several simple geometries and tested with more complex cases. The proposed network model shows excellent adaptability to arbitrary complex geometry and variable boundary conditions. The average prediction error of the network model on the testing dataset is less than 6%, and the prediction speed is two orders faster than that of the numerical simulation. In contrast to the current model, the average error of the network model with the input matrix of the binary map, traditional SDF, and model without pooling layers is around 12%, 11%, and 11%, respectively. The outstanding performance of the proposed network model indicates the potential of the deep neural network-based ROM for real-time control and rapid optimization, while encouraging further investigation to achieve practical application.

In the interdisciplinary field of data-driven models and computational fluid mechanics, the reduced-order model for flow field prediction is mainly constructed by a convolutional neural network (CNN) in recent years. However, the standard CNN is only applicable to data with Euclidean spatial structure, while data with non-Euclidean properties can only be convolved after pixelization, which usually leads to decreased data accuracy. In this work, a novel data-driven framework based on graph convolution network (GCN) is proposed to allow the convolution operator to predict fluid dynamics on non-uniform structured or unstructured mesh data. This is achieved by the fact that the graph data inherit the spatial characteristics of the mesh and by the message passing mechanism of GCN. The conversion method from the form of mesh data to graph data and the operation mechanism of GCN are clarified. Moreover, additional relevance features and weight loss function of the dataset are also investigated to improve the model performance. The model learns an end-to-end mapping between the mesh spatial features and the physical flow field. Through our studies of various cases of internal flow, it is shown that the proposed GCN-based model offers excellent adaptability to non-uniformly distributed mesh data, while also achieving a high accuracy and three-order speedup compared with numerical simulation. Our framework generalizes the graph convolution network to flow field prediction and opens the door to further extending GCN to most existing data-driven architectures of fluid dynamics in the future.

Heat convection is one of the main mechanisms of heat transfer, and it involves both heat conduction and heat transportation by fluid flow; as a result, it usually requires numerical simulation for solving heat convection problems. Although the derivation of governing equations is not difficult, the solution process can be complicated and usually requires numerical discretization and iteration of differential equations. In this paper, based on neural networks, we developed a data-driven model for an extremely fast prediction of steady-state heat convection of a hot object with an arbitrary complex geometry in a two-dimensional space. According to the governing equations, the steady-state heat convection is dominated by convection and thermal diffusion terms; thus the distribution of the physical fields would exhibit stronger correlations between adjacent points. Therefore, the proposed neural network model uses convolutional neural network (CNN) layers as the encoder and deconvolutional neural network (DCNN) layers as the decoder. Compared with a fully connected (FC) network model, the CNN-based model is good for capturing and reconstructing the spatial relationships of low-rank feature spaces, such as edge intersections, parallelism, and symmetry. Furthermore, we applied the signed distance function (SDF) as the network input for representing the problem geometry, which contains more information compared with a binary image. For displaying the strong learning and generalization ability of the proposed network model, the training dataset only contains hot objects with simple geometries: triangles, quadrilaterals, pentagons, hexagons, and dodecagons, while the testing cases use arbitrary and complex geometries. According to the study, the trained network model can accurately predict the velocity and temperature field of the problems with complex geometries, which has never been seen by the network model during the model training; and the prediction speed is two orders faster than the CFD. The ability of accurate and extremely fast prediction of the network model suggests the potential of applying reduced-order network models to the applications of real-time control and fast optimization in the future.

This study develops a geometry adaptive, physical field predictor for the combined forced and natural convection flow of a nanofluid in horizontal single or double-inner cylinder annular pipes with various inner cylinder sizes and placements based on deep learning. The predictor is built with a convolutional-deconvolutional structure, where the input is the annulus cross-section geometry and the output is the temperature and the Nusselt number for the nanofluid-filled annulus. Profiting from the proven ability of dealing with pixel-like data, the convolutional neural network (CNN)-based predictor enables an accurate end-to-end mapping from the geometry input and the desired nanofluid physical field. Taking the computational fluid dynamics (CFD) calculation as the basis of our approach, the obtained results show that the average accuracy of the predicted temperature field and the coefficient of determination R² are more than 99.9% and 0.998 accurate for single-inner cylinder nanofluid-filled annulus; while for the more complex case of double-inner cylinder, the results are still very close, higher than 99.8% and 0.99, respectively. Furthermore, the predictor takes only 0.038 s for each nanofluid field prediction, four orders of magnitude faster than the numerical simulation. The high accuracy and the fast speed estimation of the proposed predictor show the great potential of this approach to perform efficient inner cylinder configuration design and optimization for nanofluid-filled annulus.