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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. [ABSTRACT FROM AUTHOR]

This paper presents a novel reduced-order model for internal and external flow field estimations based on a sparse convolution neural network. Since traditional convolution neural network requires "rectangular" matrixes as input, the convolutional operation is computationally inefficient when applied to these problems with input matrix having sparse information. In our approach, we innovatively introduce a sparse convolution neural network (SCNN) to collect spatial information on geometries that are inherently sparse, e.g., the flow in thin pipelines in a much larger domain or the pipelines with random branches. Different from the traditional convolution neural network (CNN) model, the SCNN only collects features from areas with flow information for both the input matrix and each convolutional layer, which not only reduces the consumption of computational resources but also significantly increases network attention to flow area. The model learns the mapping relationship between geometries and the physical field of fluid flow, and the spatial positions of geometry are represented using the nearest wall signed distance function. The proposed SCNN model has the promising adaptability to arbitrary geometry and less computational resource cost compared to the traditional CNN model: the mean error of the SCNN is less than 5.4% (while the CNN is 7.1%) for the internal flow and less than 6.5% (while the CNN is 8.1%) for the external flow. Moreover, the SCNN has 72% less GPU resource usage and 52% less random access memory cost than the CNN for internal flow. For the first time, our framework introduces the sparse convolution network to flow field prediction, and the SCNN shows outstanding performance on prediction accuracy and computational resource saving for the flow problems with a sparse input information. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Numerical simulations are usually used to analyze and optimize the performance of the nanofluid-filled absorber tube with fins. However, solving partial differential equations (PDEs) repeatedly requires considerable computational cost. This study develops two deep neural network-based reduced-order models to accurately and rapidly predict the temperature field and heat flux of nanofluid-filled absorber tubes with rectangular fins, respectively. Both network models contain a convolutional path, receiving and extracting cross-sectional geometry information of the absorber tube presented by signed distance function (SDF); then, the following deconvolutional blocks or fully connected layers decode the temperature field or heat flux out from the highly encoded feature map. According to the results, the average accuracy of the temperature field prediction is higher than 99.9% and the computational speed is four orders faster than numerical simulation. For heat flux estimation, the R2 of 81 samples reaches 0.9995 and the average accuracy is higher than 99.7%. The same as the field prediction, the heat flux prediction also takes much less computational time than numerical simulation, with 0.004 s versus 393 s. In addition, the changeable learning rate strategy is applied, and the influence of learning rate and dataset size on the evolution of accuracy are investigated. According to our literature review, this is the first study to estimate the temperature field and heat flux of the outlet cross section in 3D nanofluid-filled fined absorber tubes using a deep convolutional neural network. The results of the current work verify both the high accuracy and efficiency of the proposed network model, which shows its huge potential for the fin-shape design and optimization of nanofluid-filled absorber tubes. [ABSTRACT FROM AUTHOR]

This study develops a data-driven reduced-order model based on a deep convolutional neural network (CNN) for real-time and accurate prediction of the drug trajectory and concentration field in transarterial chemoembolization therapy to assist in directing the drug to the tumor site. The convolutional and deconvoluational layers are used as the encoder and the decoder, respectively. The input of the network model is designed to contain the information of drug injection location and the blood vessel geometry and the output consists of the drug trajectory and the concentration field. We studied drug delivery in two-dimensional straight, bifurcated blood vessels and the human hepatic artery system and showed that the proposed model can quickly and accurately predict the spatial–temporal drug concentration field. For the human hepatic artery system, the most complex case, the average prediction accuracy was 99.9% compared with the CFD prediction. Further, the prediction time for each concentration field was less than 0.07 s, which is four orders faster than the corresponding CFD simulation. The high performance, accuracy and speed of the CNN model shows the potential for effectively assisting physicians in directing chemoembolization drugs to tumor-bearing segments, thus improving its efficacy in real-time. [ABSTRACT FROM AUTHOR]

A physics-informed graph convolutional network (PINN-GCN) data-driven model was proposed to solve steady-state fluid flows with various geometries. A GCN guarantees a data-driven model that adapts to irregularly structured or unstructured mesh data. In addition, the mean squared residuals of the governing equations in partial differential forms and boundary conditions are introduced into the loss function of the network model, the Physics-Informed Neural Network (PINN) increases the knowledge of the physical rules of the model, enhances the model accuracy even with sparse training data. With nine training cases containing approximately 2000 spatial point data individually, PINN-GCN could accurately predict the physical flow field over a two-dimensional object of variable size, achieving above 98.6% for mean accuracy. With five training data cases, the model could predict flow over a cylinder with variable position, achieving above 98.2% for mean accuracy. Moreover, the PINN-GCN framework can produce high-resolution predictions, even with low-resolution field training data. Furthermore, the PINN-GCN prediction speed was approximately 1000 times faster than that of the numerical simulations. Thus, the proposed model can rapidly predict high-resolution flow fields using a physics-informed, grid- and geometry-adaptive, data-driven model. • The proposed PINN-GCN model obtains the target PDEs solutions in the irregular domains with unstructured grids and without the pixelated preprocessing. • The proposed model can be trained on sparse data (low-resolution) and then predictions can be made on dense data (high-resolution). • The trained model can achieve direct predictions for domains with unseen and new geometries, which fully reflects its scalability. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• The fluid dynamic and energy equations are embedded into the neural network. • The proposed model can accurately solve natural convection with various geometries. • The model can accurately predict the physical information at the boundaries. • The model training only requires a small amount of dataset. This paper presents a novel deep learning-based surrogate model for steady-state natural convection problem with variable geometry. Traditional deep learning based surrogate models are more or less limited by the requirement for large amounts of training data, loss of accuracy due to pixelization of the original data, the low accurate prediction near the boundaries and low geometry adaptive capability. To overcome the above challenges, the proposed natural convection prediction framework is mainly composed of a physics-informed neural network (PINN) and a graph convolutional neural network (GCN), called natural convection prediction model based on physics-informed graph convolutional network (NCV-PIGN). The GCN serves as the prediction module, inferring and predicting natural convection phenomena by considering the interactions between unstructured nodes and their neighbor; the PINN incorporates the governing equations of natural convection into the loss function of the neural network, allowing the predictions from GCN to satisfy the constraints imposed by the physical laws. The advantages of this framework are twofold: the operation principles of the GCN better align with the development of the temperature field in real situations, and the embedding of physical information strengthens the model's understanding of the flow field, accurately describing the variations of temperature gradients at the boundary positions while reducing the model's reliance on training data. Finally, to validate the superiority of the NCV-PIGN, we analyze its geometric adaptability and accuracy of prediction using single and dual heat source cases. We compare the model's prediction results at different sampling point quantities and contrast them with those of purely data-driven models. The results demonstrate that the excellent geometric adaptability and prediction capability of the proposed model can be achieved with only 20 training data and once the fully trained the model can solve natural convection problems within 3 ms. The max and mean relative errors in predicting the temperature field are less than 2% and 0.4% for both single and dual heat source cases. Compared to the pure data-driven model, the proposed model has reduced the maximum error by 65.5% and the mean error by 72%. These results validate the effectiveness of the developed NCV-PIGN model, enabling better performance of the deep learning-based surrogate models for natural convection problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a neural network based reduced-order model for predicting the unsteady flow field over single/multiple cylinders. The neural network model constructs a mapping function between the temporal evolution of the pressure signal on the cylinder surface and the surrounding velocity field, where Convolutional Neural Network (CNN) layers are used as the encoder and deconvolutional neural network layers are used as the decoder. Compared with the network model with a fully connected (FC) decoder, the model with the deconvolution connected (DC) decoder is good for capturing and reconstructing the spatial relationships of low-rank feature spaces, such as edge intersections, parallelism, and symmetry, while the fluid flow, which is described by Navier–Stokes equations containing convection and diffusion terms, displays outstanding features of locality. In this article, the performance of the network models with the FC decoder and the DC decoder is evaluated by studying the problem of flow over a single cylinder first, and then the complexity of the flow structure of the studied problems is enhanced by increasing the number of cylinders and the Reynolds number. The results indicate that both the CNN-FC decoder model and CNN-DC decoder model achieve fast and accurate prediction on the velocity field, and the CNN-DC decoder model gives more robust and precise performance for all studied problems. [ABSTRACT FROM AUTHOR]

In this article, we propose an unsteady data-driven reduced order model (ROM) (surrogate model) for predicting the velocity field around an airfoil. The network model applies a convolutional neural network (CNN) as the encoder and a deconvolutional neural network (DCNN) as the decoder. The model constructs a mapping function between temporal evolution of the pressure signal on the airfoil surface and the surrounding velocity field. For improving the model performance, the input matrix is designed to further incorporate the information of the Reynolds number, the geometry of the airfoil, and the angle of attack. The DCNN works as the decoder for better reconstructing the spatial and temporal information of the features extracted by the CNN encoder. The training and testing datasets of flow fields under different conditions are obtained by solving the Navier–Stokes equations using the computational fluid dynamics method. After model training, the neural network based ROM shows accurate and dramatically fast predictions on the flow field of the testing dataset with extended angles of attack and Reynolds numbers. According to the current study, the neural network-based ROM has exhibited attractive potentials on ROM of the unsteady fluid dynamic problem, and the model can potentially serve on investigating flow control or optimization problems in the future. [ABSTRACT FROM AUTHOR]

The present study introduces a novel optimization framework that combines a Graph Convolutional Neural Network surrogate model with Genetic Algorithms (GCN-GA). This framework can optimize the heat sources layout in horizontal annuli, achieving the highest heat transfer efficiency. The predictive GCN surrogate model utilizes graph data as training input and enables it with sensitivity to geometric variations at the node level. This sensitivity allows achieving higher training and predictive accuracy compared to fully connected or convolutional neural networks. The optimization is achieved by GA with global search capabilities seeking the optimal spatial heat sources layout. The predicted average Nusselt numbers of the horizontal annuli by GCN are utilized as evaluation metrics. Simultaneously, geometric parameter features are targeted as optimization objectives, achieving optimization results consistent with numerical results across various cases. Furthermore, the GCN predicts temperature fields with average errors below 1.33% compared to numerical results. Specifically, in the cases validation involving single, double, and triple heat sources, the GCN-GA demonstrates faster optimization speeds, increased by 10, 29, and 110 times compared to numerical methods, respectively. Therefore, significant enhancements in optimization efficiency facilitated by the optimization framework built upon the GCN surrogate model. [ABSTRACT FROM AUTHOR]

Natural convection of nanofluids in annular pipes has been investigated in many studies due to its high occurrence in heat transfer systems. Solving natural convection problems numerically itself requires numerical discretization and iteration of differential equations, and introduction of nanofluids makes the solution process more complicated. In this paper, we propose and investigate a deep learning-based reduced-order model (ROM) for fast prediction of natural convection of nanofluids in annulus pipes. We construct the model using U-net structure of deep convolutional neural network and an end-to-end supervised learning method. The geometry presented by the Nearest Wall Signed Distance Function (NWSDF) is combined with numerical results to train the network model by minimizing model prediction error. The trained network model works as a ROM, which establishes the accurate mapping between the problem geometry, boundary conditions and the steady-state natural convection field. To validate the feasibility, stability and accuracy of the network model, current work studies the natural convection of nanofluids in single inner-ring annulus and multi-inner-ring annulus with different positions and diameters. The results show that the network model enables accurate predictions for natural convection of nanofluid with arbitrary inner-ring geometry, where an average accuracy of 99.9% for temperature field and 99% for velocity field are achieved; and the prediction speed of the proposed network model is more than three orders of magnitude faster than numerical solver. Furthermore, we also investigate and confirm the approach of setting the thermal boundary conditions in the neural network model, which will greatly enhance the applicability of the network model. The accurate and extremely fast prediction ability of the proposed network model proves that it can serve as a fast geometry optimization tool for design of heat exchangers or solar energy collectors of nanofluid in the future. [ABSTRACT FROM AUTHOR]