Abstract
Purpose
Current methods for flow field reconstruction mainly rely on data-driven algorithms which require an immense amount of experimental or field-measured data. Physics-informed neural network (PINN), which was proposed to encode physical laws into neural networks, is a less data-demanding approach for flow field reconstruction. However, when the fluid physics is complex, it is tricky to obtain accurate solutions under the PINN framework. This study aims to propose a physics-based data-driven approach for time-averaged flow field reconstruction which can overcome the hurdles of the above methods.
Design/methodology/approach
A multifidelity strategy leveraging PINN and a nonlinear information fusion (NIF) algorithm is proposed. Plentiful low-fidelity data are generated from the predictions of a PINN which is constructed purely using Reynold-averaged Navier–Stokes equations, while sparse high-fidelity data are obtained by field or experimental measurements. The NIF algorithm is performed to elicit a multifidelity model, which blends the nonlinear cross-correlation information between low- and high-fidelity data.
Findings
Two experimental cases are used to verify the capability and efficacy of the proposed strategy through comparison with other widely used strategies. It is revealed that the missing flow information within the whole computational domain can be favorably recovered by the proposed multifidelity strategy with use of sparse measurement/experimental data. The elicited multifidelity model inherits the underlying physics inherent in low-fidelity PINN predictions and rectifies the low-fidelity predictions over the whole computational domain. The proposed strategy is much superior to other contrastive strategies in terms of the accuracy of reconstruction.
Originality/value
In this study, a physics-informed data-driven strategy for time-averaged flow field reconstruction is proposed which extends the applicability of the PINN framework. In addition, embedding physical laws when training the multifidelity model leads to less data demand for model development compared to purely data-driven methods for flow field reconstruction.
Keywords
Citation
Rui, E.-Z., Zeng, G.-Z., Ni, Y.-Q., Chen, Z.-W. and Hao, S. (2024), "Time-averaged flow field reconstruction based on a multifidelity model using physics-informed neural network (PINN) and nonlinear information fusion", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 34 No. 1, pp. 131-149. https://doi.org/10.1108/HFF-05-2023-0239
Publisher
:Emerald Publishing Limited
Copyright © 2023, En-Ze Rui, Guang-Zhi Zeng, Yi-Qing Ni, Zheng-Wei Chen and Shuo Hao.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial & non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Investigating the flow field around structures has become a major task in wind engineering nowadays, as it directly affects the pressure distribution on structure surfaces (Li et al., 2019; Gao et al., 2021; Liu et al., 2022) and the thermal environment in their surroundings (Tian et al., 2020; Nugroho et al., 2022). Relevant studies have been carried out on investigating the flow fields around civil structures such as buildings (Flaga et al., 2018; Kim et al., 2021), bridges (Chen et al., 2016; Zhou et al., 2018; He et al., 2019) and high-speed rail tunnels (Chen et al., 2017a; Chen et al., 2017b; Li et al., 2022). Numerical simulations based on computational fluid dynamics (CFD) techniques were widely used in previous works to obtain the flow field (Liu et al., 2018; Ntinas et al., 2018; Chen et al., 2022). However, as a side effect of the high-accuracy requirements, the CFD modeling is highly time-consuming and expensive to implement (Ding et al., 2019; Wakes et al., 2021). Also, additional computational resources are needed for parametric analysis and model optimization (Penwarden et al., 2022) during the numerical simulations. Thus, an important issue in constructing flow fields is to seek a balance between accuracy and cost.
To achieve this compromise, numerous investigations have been conducted on reconstructing the flow field within the entire computational domain using a small sampling of high-fidelity data. For example, Abrahamson and Lonnes (1995) adopted the least-squares method to reconstruct the vorticity fields based on direct numerical simulation (DNS) data. Although the result of the least-squares method is close to the averaged field, it ignores details of local flow features, and, therefore, is not conducive to local characteristic analysis of the flow field. In addition, this method requires thousands of high-fidelity data points to reconstruct a flow field (Kim and Moon, 2022), which remains a heavy burden in engineering practice.
Flow field reconstruction can also be achieved by incorporating machine learning algorithms such as various neural network paradigms (Kong et al., 2021; Pruvost et al., 2001) rebuilt the complex flow features and mean velocity components in the flow field based on a radial basis function neural network using sparse experimental data. Nevertheless, as the structure and flow field become more complex, the reconstruction of flow characteristics also requires a huge cost to acquire sufficient amounts of data (Liu et al., 2011; Ladický et al., 2015; Löhner et al., 2021). In addition, as is well known, the internal flow within the flow field is controlled by the Navier–Stokes (NS) equations, which means the flow characteristics are restrained by the underlying physical rules (Sun et al., 2018; Ershkov et al., 2021). However, a purely data-driven approach for flow field reconstruction may yield results that are even contrary to some basic physical principles (Lu et al., 2021).
Machine learning, especially deep learning, has been rapidly evolving over the last couple of decades and has begun to play a predominant role across a variety of scientific disciplines (Hu et al., 2020; Eivazi et al., 2022; Jahromi et al., 2022). Due to its groundbreaking approximation capabilities in solving partial differential equations (PDEs) (Xiang et al., 2022), deep learning has become increasingly popular in the field of fluid dynamics. One of the deep learning methods which enables flow field reconstruction is the physics-informed neural network (PINN) proposed by Raissi et al. (2019). PINN recognizes the laws of physics when solving PDEs, which breaks the current deadlock of conventional machine learning methods that are confined to data-driven modeling (Lu et al., 2021; Sun et al., 2021). In other words, PINN frees machine learning methods from their dependence on a large demand for experimental or field-measured data (Rao et al., 2020), thus, offering a promising alternative to solving real-world problems (Rui et al., 2023; Hasanuzzaman et al., 2023). In addition, as aforementioned, extrapolation or observation bias may lead to poor generalization performance for purely data-driven models, and as a result, predictions by such models may not be physically consistent. PINN, on the other hand, uses the residuals of the physical governing equations to form a loss function for neural network training, which serves as a penalty to restrict the space of feasible solutions. PINN can also combine traditional physical models with sparse high-fidelity measurements to reconstruct flow fields (Karpatne et al., 2017; Eivazi and Vinuesa, 2022), which has been a research hotspot and attracted tremendous attention in recent years (Wang et al., 2021; Wang et al., 2022). For instance, Arzani et al. (2021) applied the PINN strategy to near-wall blood flow reconstruction, which incorporated the fluid governing functions and sparse internal data into the loss function. The numerical results demonstrated the consistency of PINN predictions with the results from conventional CFD methods. Jin et al. (2021) established PINN models to reconstruct laminar and turbulent flows based on NS equations and sparse velocity data on domain boundaries. These PINN-related investigations demonstrate its ability in flow field reconstruction, as well as its generalization capability with fewer training data while remaining accurate. Nevertheless, the limitations of this approach also result in an impediment to the development of its engineering applications, including computational cost and difficulty in dealing with complex problems (Goswami et al., 2020).
To overcome the disadvantages of the above methods, a multifidelity physics-informed data-driven strategy for time-averaged flow field reconstruction is proposed in this article. First, an approximate low-fidelity flow field is obtained from the PINN prediction, which uses Reynold-averaged NS (RANS) equation as the governing function. The RANS equation is the time-averaged form of the NS equation. Central to it is Reynolds decomposition, which separates a transient flow quantity into a time-averaged component and a fluctuating component. It transforms the unsteady turbulent problem into a steady problem, which greatly reduces the computational cost. In engineering practice, the RANS equation has already become the most commonly used fluid governing equation. Second, sparse field or experimental measurements are used as high-fidelity observations. The nonlinear information fusion (NIF) algorithm proposed by Perdikaris et al. (2017) is then adopted to establish a multifidelity Gaussian process (GP) model for flow field reconstruction. The nonlinear cross-correlations between low-fidelity approximations and high-fidelity observations are extracted to train the multifidelity GP model and make high-fidelity predictions using the NIF algorithm. Two case studies regarding time-averaged flow field reconstruction are presented to verify the feasibility of the proposed method, including a flow past a hill and a flow past a square cylinder. We also compare the performance of the proposed strategy with other commonly used methods. The results demonstrate that the multifidelity model has superior accuracy in approximating measurement data in the two cases.
The proposed strategy greatly extends the applicability of the physics-based PINN framework, where the PINN model is only used for low-fidelity modeling which requires less accuracy. Furthermore, the embedded physical laws provide essential guidance in multifidelity modeling which results in less data demand in flow field reconstruction compared with purely data-driven methods. The remainder of this article is organized as follows. Section 2 presents the PINN methodology for solving RANS equations and the NIF algorithm for multifidelity modeling. In Section 3, we demonstrate the performance of our proposed strategy for flow field reconstruction in two two-dimensional (2 D) turbulent flow cases. Main conclusions are drawn in Section 4.
2. Methodology
2.1 Physics-informed neural network for solving Reynold-averaged Navier–Stokes equations
In this study, PINN is constructed to yield approximate solutions to the 2D RANS equations. The RANS equations for simulating turbulent flows are expressed as:
PINN is a kind of deep neural network that takes the residual of physical constraints as the loss function. During the calculation, there will be thousands of collocation points scattered inside or on the boundaries of the computational domain of the governing equations, which are used to compute the residuals of the governing equations and boundary conditions. These residuals are further embedded in the loss function of the neural network, and they tend to converge toward zero during the training process with the aid of an optimizer. The schematic diagram of PINN that we construct for flow field reconstruction is depicted in Figure 1. The left part of the PINN is a fully connected neural network that maps the relationship between spatial coordinates (x, y) and the characteristics of the flow field (φ, p), here φ represents the stream function which contains the velocity information, while p represents the pressure.
It is noteworthy that here we adopt stream function instead of velocity to force the fluid continuity equation as a hard constraint. The relationship between the stream function and fluid velocity components is described as:
where:
In the above expressions, Lf and Lb denote the loss components corresponding to the residuals of the governing equations and boundary conditions, respectively; wf and wb denote the weighting coefficients of the corresponding loss terms;
2.2 Nonlinear information fusion algorithm for multifidelity modeling
The NIF algorithm proposed by Perdikaris et al. (2017) enables us to combine low-fidelity models with small amounts of high-fidelity observations for multifidelity modeling, which mainly relies on the principled framework of GP regression. Here we define fh and fl as the GPs that model the data on the high and low-fidelity levels, respectively. In the NIF algorithm, fh is expressed in the following form:
Here,
The hyperparameters are optimized by minimizing the negative log marginal likelihood (NLML) of the GP model, which is described as:
Notice that Monte Carlo simulation is used here to obtain the posterior distribution of the high-fidelity model. This is because only the low-fidelity model is a standard GP regression with parametric input data points, and its posterior prediction follows a Gaussian distribution. Whereas the high-fidelity model is a GP regression model with the input of the posterior prediction from the low-fidelity model. As a result, the posterior distribution of the high-fidelity model is no longer Gaussian. In view of this, Monte Carlo integration is pursued on equation (11) to calculate the posterior mean and variance of the high-fidelity model.
2.3 Workflow of the proposed strategy
The workflow of the proposed multifidelity flow field reconstruction method is summarized as follows:
Step 1: A PINN is trained using the RANS equations. The residuals of the governing equations and boundary conditions at sampled collocation points are embedded in the loss function to train the PINN model.
Step 2: An immense amount of labeled data, needed for low-fidelity GP modeling, is generated from the PINN model within the computational domain. The hyperparameters in the low-fidelity GP model are optimized by minimizing the NLML of equation (10). At last, the posterior mean and variance of the low-fidelity standard GP regression model are calculated.
Step 3: The high-fidelity GP regression model in equation (8) is established based on the posterior prediction of the low-fidelity model and small amounts of high-fidelity observations. The hyperparameters of the high-fidelity GP model are optimized by minimizing the NLML of equation (10), which uses the kernel function of equation (9).
Step 4: The posterior mean and variance of the high-fidelity GP model are evaluated by Monte Carlo integration on equation (11), which uses the posterior mean and variance of the low-fidelity standard GP regression model obtained in Step 2.
3. Results and discussions
3.1 Case 1: flow past a hill (Reynolds number: 6 × 104)
The data used in the first case study, which are open to all fluid dynamicists around the world, were obtained by Almeida et al. (1993) in an experimental study. In the experiment, a fully developed channel flow passed through a single hill at the location of 6 m along the flow direction from the tunnel inlet. Meanwhile, the time-averaged flow velocities in both horizontal and vertical directions around the hill were measured, which will serve as a test database to verify the feasibility of our proposed multifidelity flow field reconstruction method. In this case study, we aim to reconstruct the mainstream velocity of the channel flow by using the low-fidelity PINN predictions and the high-fidelity experimental measurements under the multifidelity modeling strategy.
Making use of the RANS equations and boundary conditions, we first formulate a PINN which can offer solution to the two-dimensional (2D) time-averaged flow field around the hill. It can be seen in equation (2) that the introduction of the Reynolds stress terms makes the RANS equations no longer a closed-form system of equations. To close the RANS equations, Chen’s model (Chen and Xu, 1998) is adopted which assumes the Reynolds stress terms can be expressed as:
More specifically, the residuals of the Chen’s model-based RANS equations calculated at the 9,766 domain collocation points form the loss term Lf defined in equation (6), and the residuals of the boundary conditions calculated at the 2,000 boundary collocation points form the loss term Lb defined in equation (7). By minimizing these physical constraints, the configured PINN realizes its function of offering approximate solutions to this flow problem. The PINN predictions of the velocity component u after 1 × 105 training iterations are shown in Figure 3(a), which are depicted as an orange-curved surface, compared with the red dots which represent the experiment measurements. Figure 3(b) depicts the contour of the velocity component u based on the PINN prediction, which is also compared with the reference data from the experiment. As can be seen in Figure 3, there exists a significant difference between the PINN predictions and the experimental results.
We can easily draw a conclusion from Figure 3 that, without incorporating measurement data to train the PINN, its solution can only be viewed as low-fidelity approximation. To establish the proposed multifidelity model for predicting flow field around the 2D hill, we then select 900 uniformly distributed low-fidelity sampling points (labeled data) generated by the PINN prediction. Meanwhile, among the total 325 experimental measurement points scattered in the computational domain, we pick up 35 points and consider the measured mainstream velocities at these points as high-fidelity training data. The spatial coordinates of the 35 high-fidelity training points are shown in Table 1. In selecting the training points, we abide by the principle of distributing the training points over the whole computational domain as evenly as possible.
Then, the NIF algorithm is implemented to establish the multifidelity model. We first train a GP regression model by using the low-fidelity data to acquire the Gaussian predictive posterior distribution of the mainstream velocity component u on the low-fidelity level. We maximize the marginal log-likelihood to seek optimal hyperparameters by using L-BFGS optimizer with the randomized restart strategy. Once acquiring the Gaussian posterior distribution on the low-fidelity level, we proceed to formulating the high-fidelity GP regression model according to equation (8). The negative marginal log-likelihood defined in equation (10) on the high-fidelity level is maximized to optimize the hyperparameters. After the model is fully trained, we get the posterior distribution of the mainstream velocity component u on the high-fidelity level using Monte Carlo integration. The prediction results of the velocity component u using the proposed multifidelity model are first shown in Figure 4(a). In this figure, the green-curved surface represents the prediction of the mainstream velocity component u obtained by the multifidelity model. The yellow curved surface denotes the low-fidelity data generated by the PINN prediction. The red dots are the experimental results of the mainstream velocity component u at all measurement points. In general, the low-fidelity data (PINN predictions) and the predictions by the multifidelity model show a similar trend within the whole computational domain; however, the latter is much closer to the experimental results. This is because the NIF algorithm can precisely capture the nonlinear nonfunctional space-dependent cross-correlations between the low-fidelity and high-fidelity data. As a result, the multifidelity model can learn from the trend of the low-fidelity PINN predictions to fit the scattered data points on the high-fidelity level. In other words, scattered (sparse) data points on the high-fidelity level are used to correct the low-fidelity prediction surface, while the trend of the low-fidelity surface stemming from the physical law is preserved to the greatest extent. The velocity contours of u from the multifidelity model predictions and from the experimental results are compared in Figure 4(b). The absolute error is much lower than that of the low-fidelity model which is illustrated in Figure 3(b).
Figure 5 provides a more detailed comparison of the results between the multifidelity model predictions and the high/low-fidelity data on 12 vertical lines within the computational domain. We can easily discover that the multifidelity model predictions show good consistency with the experimental measurements in the whole computational domain. Compared with other lines, better results are achieved on the lines x = −0.050 m, x = 0.050 m, x = 0.150 m, x = 0.300 m and x = 0.500 m because the 35 high-fidelity training data points are positioned there. On the other lines, the multifidelity model yet demonstrates competitive results, especially when being compared with the PINN predictions. We further quantitatively evaluate the performance of the proposed multifidelity model in Table 2, through comparison with other two widely used strategies for flow field reconstruction, i.e. data-driven PINN and conventional perceptron neural network (PNN). In the data-driven PINN strategy, we not only embed the physical governing equations and boundary conditions into the total loss but also concurrently embed the 35 high-fidelity training data (Table 1) into the PINN training process. For consistency, the configuration of the data-driven PINN is the same as that of the PINN we used to train the low-fidelity model. In the PNN paradigm, we simply conduct a regression task using only 35 high-fidelity training data points. A PNN configuration with only one hidden layer which contains 10 neurons is adopted, while the L-BFGS optimizer with a learning rate of 5 × 10−4 is adopted to train the neural network. As shown in Table 2, we use the ℓ2 error to quantify the prediction accuracy, which is defined as:
3.2 Case 2: flow past a square cylinder (Reynolds number: 2 × 104)
The data used in the second case study are from the experiment conducted by Lyn and Rodi (1994), which describe a turbulent flow around a 2D square cylinder, as shown in Figure 6. The computational domain is 0.44 m in length and 0.32 m in width, while a 0.04 m × 0.04 m square cylinder is located at the left center of the computational domain. Point A is the bottom left corner of the square cylinder, whose spatial coordinate is (0.10, 0.14) as shown in Figure 6. The left boundary of the computational domain is defined as an initial speed boundary, where the fluid velocity stabilizes at 0.535 m/s. The upper and lower boundaries are defined as symmetry boundaries, while the right boundary of the computational domain is assigned a zero-pressure outlet. The surfaces of the square cylinder are considered as wall boundaries, where the fluid velocity equals zero. In the experiment, there were 517 measurement points that captured the time-averaged flow velocity components u and v inside the flow field.
Case 2 differs from Case 1 in two aspects. First, the most intuitive aspect is the different geometric appearances. Second, for comparison with Case 1, both the components v and u at selected measurement points are used to formulate the multifidelity model in Case 2. Among the 517 experimental measurement points, we pick up 36 points scattered on 9 individual vertical lines to train the multifidelity model for time-averaged flow field reconstruction. Their spatial coordinates are shown in Table 3. In selecting training points, we still abide by the principle of distributing the training points over the whole computational domain as evenly as possible.
The objective of Case 2 remains to apply all available physical restrictions and sparse measurement information to reconstruct high-fidelity mainstream velocity u over the entire computational domain. In Case 2, we compare the performances of five schemes to achieve this objective. In the first three schemes, the NIF algorithm and sparse u measurements are used in training the multifidelity model. The only difference between these three schemes is the low-fidelity data source. A PINN without training data (purely physics-based), a PINN with v embedded in its training process and CFD are used as the low-fidelity data sources, respectively. The fourth scheme is a data-driven PINN where both u and v are used to train the neural network, while the NIF algorithm is not engaged in this scheme. The fifth scheme is a PNN like what was built in Case 1. The five schemes are described in Table 4.
Considering the multiple sources of low-fidelity data in this case study, we delve further into the details of the PINN and CFD frameworks separately. For the PINN framework, its configuration remains the same as that in the previous case except that only 108 collocation points inside the 2 D square cylinder are excluded, which ultimately leads to a total of 9,892 collocation points inside the domain. In addition, there are 50 equally-spaced boundary collocation points on the initial speed boundary, zero-pressure outlet, upper symmetry boundary, lower symmetry boundary and each of the four side surfaces of the 2D square cylinder, respectively. Thus, a total of 400 boundary collocation points are used to calculate the physical residuals in this case. For the CFD framework, the simulation of the time-averaged flow field is performed based on the commercial software Star CCM+ in this study, and the mesh inside the computational domain consists of more than 36,484 cells. For both PINN and CFD frameworks, the standard k–ε model is adopted in RANS turbulence modeling. In the standard k–ε model, the Reynolds stress is described as follows according to the Boussinesq assumption in 2 D cases:
The prediction results of the mainstream velocity u from different schemes are depicted in Figure 7, which are also compared with the experiment results. These velocity contours show that Scheme 2, notably in the upstream region of the square cylinder, reconstructs the fluid features most accurately and efficiently. This may be due to the fusion of measured fluid features from the upstream regions in low-fidelity modeling in this scheme. Again, ℓ2 error is used to quantitatively evaluate the accuracy of predictions from different models. As shown in Table 6, the ℓ2 error of Scheme 2 is the lowest among all the schemes, which is only 8.8% compared with the experimental results. In particular, when v-embedded PINN, instead of CFD, is used as the low-fidelity data source, the ℓ2 error decreases by 6.6%. This is because measurement information has been included; compared to CFD, the simulation results from v-embedded PINN are more accurate. However, when field information is unavailable, the precision of the PINN’s prediction drops dramatically. This results in the ℓ2 error of Scheme 1 being 18.1%, which is the worst among the multifidelity models. We observe the significance of physical information fusion in neural network modeling, which results in an improved accuracy by 13.6% for Scheme 4 over Scheme 5. Meanwhile, it is worth mentioning that the modeling process in all PINN-involved schemes (i.e. Schemes 1, 2 and 4) is time-consuming, compared to CFD, because of involving the training of PINN. However, once the model has been fully trained, it can quickly predict wind velocity at any point within the computational domain. In the multifidelity strategy, PINN-based low-fidelity models can be fully trained off-line. Thus, its training cost is not a critical concern. Once we obtain measurement data, the multifidelity modeling can be completed in just a few seconds.
4. Conclusions
In this paper, we proposed a novel time-averaged flow field reconstruction strategy in the framework of multifidelity modeling using PINN and the NIF algorithm. It can be viewed as a two-step approach in which the low-fidelity data are generated by a PINN, while sparse experimental or field measurement data are thought of as high-fidelity data which, in conjunction with PINN-generated low-fidelity data, are used to formulate a multifidelity model by means of the NIF algorithm. A flow past a hill and a flow past a square cylinder were used to verify the capability of the proposed multifidelity strategy, and the results demonstrated its efficacy for time-averaged flow field reconstruction. This study comes to the following conclusions:
Although only small amounts of measurement/experimental data are accessible for reconstructing the time-averaged flow fields in both cases, the missing flow information within the whole computational domain can be favorably recovered by the multifidelity model.
The predictions of PINN constructed purely using the RANS equations can only be considered as low-fidelity data due to low prediction accuracy in both cases. However, its trend (underlying physics) within the whole computational domain can be inherited through PINN, while the high-fidelity yet sparse measurement data can be used to rectify the low-fidelity prediction surface by implementing the NIF algorithm. The multifidelity model elicited by the NIF algorithm can learn from the trend of the low-fidelity PINN predictions to fit the scattered data points on the high-fidelity level, thereby enabling the reconstruction of time-averaged flow fields with high accuracy.
Compared with other flow field reconstruction strategies, the proposed strategy demonstrates the most competitive results. The relative errors of the mainstream velocity component from the multifidelity prediction are less than 10% in both cases relative to the experimental measurements. However, the modeling process in all PINN-involved strategies is time-consuming because of involving the training of PINN. Once the model has been trained, it can make predictions quickly.
Figures
Figure 1.
PINN framework for flow field reconstruction
Figure 2.
Computational domain of the flow past a 2D hill
Figure 3.
Prediction of u using the low-fidelity model in Case 1: (a) general view; and (b) the velocity contour compared with the experimental counterpart (absolute error = prediction − experimental result)
Figure 4.
Prediction of u using the multifidelity model in Case 1: (a) general view; and (b) the velocity contour compared with the experimental counterpart (absolute error = prediction − experimental result)
Figure 5.
Comparison of the results between the multifidelity model predictions and the high/low-fidelity data on 12 vertical lines in Case 1
Figure 6.
Computational domain of the flow passing a 2D square cylinder
Figure 7.
Prediction of u using five schemes: (top) prediction; (middle) experimental result; and (bottom) absolute error
Spatial coordinates of the high-fidelity points used to formulate the multifidelity model
| x Coordinate (m) | −0.050 | 0.050 | 0.150 | 0.300 | 0.500 |
|---|---|---|---|---|---|
| y Coordinate (m) | 0.006 | 0.002 | 0.001 | 0.001 | 0.001 |
| 0.015 | 0.015 | 0.015 | 0.016 | 0.016 | |
| 0.030 | 0.030 | 0.030 | 0.030 | 0.030 | |
| 0.070 | 0.070 | 0.070 | 0.070 | 0.070 | |
| 0.100 | 0.100 | 0.100 | 0.100 | 0.100 | |
| 0.130 | 0.130 | 0.130 | 0.130 | 0.130 | |
| 0.160 | 0.165 | 0.165 | 0.165 | 0.165 |
Source: Table by authors
Performance of different flow field reconstruction strategies
| Multifidelity model | Data-driven PINN | PNN | |
|---|---|---|---|
| ℓ2 Error | 9.8% | 24.6% | 55.3% |
| Computing time (s) | 2.3 × 104 | 2.3 × 104 | 4.1 × 101 |
Source: Table by authors
Spatial coordinates of the high-fidelity points used to train the multifidelity model
| x Coordinate (m) | 0.000 | 0.100 | 0.155 | 0.200 | 0.250 | 0.300 | 0.350 | 0.392 | 0.400 |
|---|---|---|---|---|---|---|---|---|---|
| y Coordinate (m) | 0.200 | 0.200 | 0.160 | 0.160 | 0.160 | 0.160 | 0.160 | 0.160 | 0.160 |
| 0.240 | 0.240 | 0.180 | 0.200 | 0.200 | 0.200 | 0.200 | 0.200 | 0.200 | |
| 0.280 | 0.280 | 0.200 | 0.240 | 0.240 | 0.240 | 0.240 | 0.240 | 0.240 | |
| 0.320 | 0.320 | 0.220 | – | – | 0.280 | – | 0.280 | 0.280 | |
| – | – | – | – | – | 0.320 | – | 0.320 | 0.320 |
Source: Table by authors
Five schemes for flow field reconstruction in Case 2
| Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | Scheme 5 |
|---|---|---|---|---|
| PINN + NIF | v-embedded PINN + NIF | CFD + NIF | u- and v-embedded PINN | PNN |
Source: Table by authors
The values of the coefficients in the standard k–ε model
| Coefficient | Cμ | C1ε | C2ε | σk | σε |
|---|---|---|---|---|---|
| Value | 0.09 | 1.44 | 1.92 | 1.0 | 1.3 |
Source: Launder and Sharma (1974)
Performance of different flow field reconstruction strategies
| Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | Scheme 5 | |
|---|---|---|---|---|---|
| ℓ2 Error | 18.1% | 8.8% | 15.4% | 15.5% | 29.1% |
| Computing time (s) | 2.8 × 104 | 2.8 × 104 | 1.8 × 103 | 2.8 × 104 | 2.1 × 101 |
Source: Table by authors
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Acknowledgements
The work was supported by a grant from the Research Grants Council (RGC) of the Hong Kong Special Administrative Region (SAR), China (Grant No. PolyU 152308/22E) and grants from The Hong Kong Polytechnic University (Grant No. 1-WZ0C, 1-BD23). The authors also appreciate the funding support by the Innovation and Technology Commission of Hong Kong SAR Government to the Hong Kong Branch of National Engineering Research Center on Rail Transit Electrification and Automation (Grant No. K-BBY1).
Data availability statement: The data that support the findings of this study are available from the corresponding author upon reasonable request.
Declaration of conflict of interest: The authors declare that there is no conflict of interest.