Research on a Rapid Three-Dimensional Compressor Flow Field Prediction Method Integrating U-Net and Physics-Informed Neural Networks

Abstract

This paper presents a neural network model, PINN-AeroFlow-U, for reconstructing full-field aerodynamic quantities around three-dimensional compressor blades, including regions near the wall. This model is based on structured CFD training data and physics-informed loss functions and is proposed for direct 3D compressor flow prediction. It maps flow data from the physical domain to a uniform computational domain and employs a U-Net-based neural network capable of capturing the sharp local transitions induced by fluid acceleration near the blade leading edge, as well as learning flow features associated with internal boundaries (e.g., the wall boundary). The inputs to PINN-AeroFlow-U are the flow-field coordinate data from high-fidelity multi-geometry blade solutions, the 3D blade geometry, and the first-order metric coefficients obtained via mesh transformation. Its outputs include the pressure field, temperature field, and velocity vector field within the blade passage. To enhance physical interpretability, the network’s loss function incorporates both the Euler equations and gradient constraints. PINN-AeroFlow-U achieves prediction errors of 1.063% for the pressure field and 2.02% for the velocity field, demonstrating high accuracy.

1. Introduction

The development of the aviation industry places higher demands on the rapid iterative design of aeroengines. To achieve higher compressor pressure ratios and efficiency, continuous optimization of compressor design is necessary. Accurate prediction of the internal flow characteristics of compressors, especially the fluid acceleration at the blade leading edge and the flow characteristics of the wall boundary layer, which are critical for understanding phenomena like laminar–turbulent transition in gas turbine engines [1], is crucial for the rapid iterative design of compressors. These phenomena often involve complex turbulence, which remains a significant challenge in CFD [2]. Although traditional Computational Fluid Dynamics (CFD) methods can obtain accurate solutions of the flow field through numerical simulation, their computational speed no longer meets the requirements of modern compressor design [3]. This is because thousands of evaluations are often required during optimization, and each high-fidelity 3D simulation may take hours or even days to complete.

Deep learning-based methods provide a new perspective for compressor blade flow field prediction. Deep learning models can infer potential physical laws from large amounts of data, especially showing potential in handling high-dimensional and nonlinear problems [4,5]. The latest advancements include Transformer-based architectures, which are increasingly being applied to complex fluid dynamics problems [6]. Lv Zhaoyang et al. [7] used a convolutional neural network (CNN) to build a prediction model for airfoil flow fields, with the input being the original image of the airfoil flow field represented by pixels and the output being the lift coefficient and moment coefficient of the airfoil. The experimental results showed that the CNN model’s predictions had good accuracy and the calculation speed was 40 times faster than CFD methods. N. Thuerey et al. [8] designed a U-Net-based network [9] that, trained on a large amount of CFD data, could predict the velocity and pressure distribution of airfoil flow fields from boundary conditions, with an average prediction error of less than 3%. S. Bhatnagar et al. [10] introduced Gradient Sharpening (GS) to improve the clarity of predictions and used a CNN structure with shared encoding and decoding layers to significantly improve computational efficiency.

However, in fluid dynamics applications, deep learning models face specific challenges. For example, most deep learning models commonly adopt a pixelated preprocessing method that projects data onto a uniform Cartesian grid, which leads to the loss of flow field information in small-scale regions near the wall and boundary, affecting prediction accuracy. While coordinate transformation [11,12,13] can mitigate this issue, emerging implicit neural representations [14] offer a mesh-independent continuous flow field prediction method. To capture the flow field information near the wall, Z. Yang et al. [15] proposed the AMGNET method, which relies on algebraic multigrid algorithms and graph neural networks (GNNs) to improve the accuracy of flow field prediction in small-scale regions. However, although graph neural networks can handle unstructured grids, they have problems in representational theory and limitations in hardware requirements, parallel processing, and distributed solving.

Compared to well-structured text and images, graphs are irregular. A fundamental assumption in machine learning on graphs is that the prediction target should be independent of the order of nodes on the graph. To satisfy this assumption, GNNs introduce an inductive bias called permutation invariance [16]. Specifically, the output of GNNs is independent of how the node indices of the graph are assigned and their processing order, i.e., the model parameters are independent of the node order and are shared across the entire graph. Due to the unique parameter sharing mechanism of GNNs, the way they fit data features is very different from traditional neural networks. First, most GNNs are usually used as black-box feature extractors for graphs, and it is unclear how well they can capture different graph features and topologies. Second, due to the introduction of permutation invariance, the results of the classical universal approximation theorem for neural networks (NNs) [17,18] cannot be directly generalized to GNNs [19]. Furthermore, the limitations of the expressive power of GNNs are related to some longstanding difficult problems in graph theory [20]. For example, in predicting the properties of chemical molecules, it is necessary to determine whether the molecular structure is the same as or similar to molecules with known properties, which involves problems such as graph/subgraph isomorphism judgment [21,22].

Since existing deep learning models cannot overcome the problem of information loss in small-scale regions caused by structured grids, researchers have proposed methods for grid transformation in the preprocessing stage. L.-W. Chen and N. Thuerey [11] improved the solution efficiency and maintained the ability to capture complex flow phenomena by using a U-Net network to transform unstructured grids into body-fitted Cartesian grids. W. Cao et al. [12] combined physics-informed neural networks [23] with grid transformation and achieved a reduction in error of nearly one order of magnitude compared to the second-order Finite Volume Method (FVM) on very sparse grids, solving the problem of local sharp transitions in airfoil flow fields. J. Hu and W. Zhang [13] also used grid transformation, retaining the geometric information of the fluid equations, and the prediction accuracy of the model with preprocessed input data was significantly higher than the prediction results of Principal Component Analysis (PCA). T. Zhao et al. [24] combined deep convolutional neural networks with a physics-constrained method based on the Euler equations, significantly reducing the dependence on the amount of training data and making the training process more stable. This approach is particularly relevant for complex flow regimes, including transonic flows [25]. Further advancements in physics-informed neural networks, such as cascade loss models, continue to enhance their capabilities [26].

Current research on deep learning-based flow field reconstruction mainly focuses on two-dimensional flow fields. However, in complex scenarios, two-dimensional flow fields lack crucial information in one dimension, and only three-dimensional flow fields can meet research needs, such as the airflow around vehicles, the aerodynamics of aircraft wings, and the rotating stall of compressor blade rows. To convert a three-dimensional flow field into a two-dimensional flow field for model input, it is usually necessary to perform slicing processing. This not only reduces the intuitiveness of flow field features but also requires a cumbersome preprocessing step [27]. Moreover, low-cost storage of large-scale three-dimensional simulation field data is more attractive compared to storing two-dimensional flow field data. Furthermore, the transition from studying two-dimensional flow fields to three-dimensional flow fields helps to further research and apply deep learning techniques in the field of fluid dynamics [28]. For example, Bruni et al. [29] have successfully applied deep learning to multi-stage axial compressors, demonstrating its potential for rapid design iteration and analysis in complex turbomachinery. Therefore, three-dimensional models developed for three-dimensional input are more conducive to engineering applications such as real-time fluid control, field analysis, and online prediction tasks.

In this study, we propose an end-to-end, data-driven framework for three-dimensional compressor flow field prediction that maps blade geometry and freestream conditions directly to three-dimensional steady-state flow field distributions. Unlike existing approaches that rely on intermediate surrogate modeling, dimensionality reduction, or scalar output regression (e.g., lift or loss coefficients), our proposed method performs full-field prediction without intermediate reconstruction steps, enabling a more direct and physically comprehensive mapping from design parameters to flow solutions. This framework integrates coordinate transformation, U-Net-based encoding–decoding, and physics-informed loss constraints. While each component exists in prior works, our approach introduces several key improvements that significantly enhance prediction performance and physical fidelity in compressor applications, including the integration of attention mechanisms [30] and advanced network components: (1) The model performs direct 3D flow field prediction without requiring 2D slicing or flow reconstruction, thereby preserving the spatial coherence and reducing preprocessing complexity. (2) We adopt a variable-specific decoder structure with separate optimization paths for pressure, temperature, and velocity components, improving the treatment of multi-scale physical interactions. (3) The coordinate transformation module embeds first-order metric coefficients, and the training is guided by Euler-based residual loss, enabling accurate resolution of shocks, boundary layers, and wake structures in highly curved blade passages. These innovations enhance both the efficiency and interpretability of flow predictions, offering practical advantages for iterative compressor design and real-time aerodynamic analysis.

2. Materials and Methods

To establish a physically grounded predictive model, we first define the configuration of the studied system. The object of study is a subsonic compressor blade operating under steady flow conditions.

Figure 1 illustrates the chordwise shape of the CDA (Controlled Diffusion Airfoil) blade. The curved shape represents the cross-section of a blade, with A and B marking the leading and trailing edges. The straight line connecting A and B is the chord line, and its length is the chord length (b), consistent with Figure 1. The angle $\theta$ again denotes the stagger angle or blade setting angle, specifying the airfoil’s orientation. The internal circles, often used conceptually, illustrate the concepts of camber and thickness, which define the airfoil’s shape. $f_{\max }$ represents the maximum camber, which is the greatest perpendicular distance from the chord line to the mean camber line (the line equidistant from the upper and lower surfaces). Its longitudinal position from the leading edge along the chord is denoted by a. Similarly, $c_{\max }$ is the maximum thickness of the airfoil, the greatest perpendicular distance between the upper and lower surfaces, and its axial location from the leading edge is indicated by e. In essence, this figure elucidates the fundamental parameters of an airfoil’s cross section, which directly influence its aerodynamic performance, such as lift generation and drag.

Figure 2 presents a two-dimensional representation of a linear blade cascade, typical in turbomachinery components such as compressors and turbines. It comprehensively displays the flow path through a single blade passage along with its associated velocity triangles and key geometric parameters. Lines 1-1 and 2-2 define the inlet and outlet flow sections, often parallel for linear cascades, indicating uniform flow conditions upstream and downstream. The curved lines represent the airfoil (blade) profiles, with points A and B denoting the leading and trailing edges, respectively. Geometrically, t signifies the pitch of the cascade, which is the perpendicular distance between corresponding points on adjacent blades, while b is the chord length, the straight-line distance between the leading and trailing edges of a single airfoil. From an aerodynamic perspective, $w_1$ and $w_2$ denote the relative flow velocities at the inlet and outlet of the cascade, respectively. The angles ${\beta }_1$ and ${\beta }_2$ are the inlet and outlet relative flow angles, measured between the relative velocity vectors and the axial direction (lines 1-1 and 2-2). The blade angles at the leading and trailing edges are represented by ${\beta }_{1K}$ and ${\beta }_{2K}$, respectively. The incidence angle (i) is defined as the difference between the inlet blade angle and the inlet flow angle ($i={\beta }_{1K}-{\beta }_1$), indicating how directly the flow hits the leading edge. Conversely, the deviation angle ($\delta$) is the difference between the actual outlet flow angle and the outlet blade angle ($\delta ={\beta }_2-{\beta }_{2K}$), reflecting the flow’s inability to perfectly align with the blade surface at the exit. The stagger angle ($\theta$) defines the orientation of the entire blade, measured between the chord line (AB) and the axial direction. Additional angles, ${\beta }_m$ and ${\beta }_y$, likely represent mean or geometric blade angles at the leading and trailing edges, respectively, often related to the mean camber line of the blade. The velocity triangle on the right, involving $w_m$ (a mean velocity) and $w_t$ (a tangential velocity component), provides a graphical depiction of the vector relationships between different velocity components and angles within the cascade, crucial for turbomachine performance analysis.

The overall workflow of the proposed prediction model is illustrated in Figure 3 and consists of four main stages: (1) CFD data preparation: We begin by generating a structured dataset of 3D steady-state CFD flow fields using a set of parametric compressor airfoils under varying inflow conditions. Each sample includes flow variables and geometric metadata. (2) Coordinate transformation (CoordTrans): The raw 3D spatial coordinates are passed through a geometry-aware transformation module, which normalizes the mesh and embeds relative geometric features into the input tensor. This module outputs transformed spatial information and blade-specific metrics (see “CoordTrans” in Figure 3). (3) Neural network inference: The transformed data, combined with global flow metadata, is fed into a U-Net-based 3D convolutional neural network (AeroFlow-U or PINN-AeroFlow-U), which predicts the full-field distribution of pressure, temperature, and velocity components. (4) Loss computation and training: During training, the model minimizes a composite loss function, including pointwise data loss, gradient consistency loss, and a physics-informed residual term based on the Euler equations. The training process is performed offline, while inference is nearly instantaneous. These four stages are visually mapped in Figure 3, which provides a schematic overview of the full learning pipeline.

2.1. Research Objectives

This study aims to construct a flow field prediction model, PINN-AeroFlow-U, based on the U-Net network architecture and incorporating physics-informed neural networks (PINNs). In the data preprocessing stage, the flow field data is mapped from the non-uniformly distributed physical space to a uniformly discrete computational space, enhancing the model’s ability to represent the complex features of the compressor flow field. In the model prediction output stage, a PINN is introduced to embed the Euler equations as constraints into the model’s loss function, further improving the model’s adherence to physical laws and prediction accuracy.

The object of this study is the subsonic compressor flow field. Under inviscid conditions, the Reynolds-Averaged Navier–Stokes (RANS) equations can be simplified to the Euler equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \mathbf{u}\right)=0,\end{array}$$

(1)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}\right)& =-\nabla p+\rho \mathbf{g},\end{array}$$

(2)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial E}{\partial t}}+\nabla ·\left((E+p)\mathbf{u}\right)& =-\mathbf{u}·\nabla p+Q.\end{array}$$

(3)

In the above equations, $\rho$ is the fluid density, $\mathbf{u}$ is the fluid velocity vector, p is the fluid pressure, $\mathbf{g}$ is the body force (such as gravity), E is the total energy of the fluid, and Q is the heat source term per unit volume.

In this paper, the initial geometry and inlet conditions I, grid coordinate information N, deep learning model f, and flow field variables F and their relationships are defined as

$$\begin{array}{cc}\hfill I& =\left[M{a}_{\infty },Ao{A}_{\infty },C_{max},\theta ,\beta \right],\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill F& =[p,\rho ,t,u,v,w],\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill F& =f\left(I,N\right).\hfill \end{array}$$

(6)

In Equation (6), $M{a}_{\infty }$ represents the airflow Mach number at the inlet, $Ao{A}_{\infty }$ represents the angle of attack, $C_{max}$ represents the maximum relative thickness, $\theta$ represents the blade camber angle, and $\beta$ represents the airflow inlet angle.

The process of solving the model can be defined as optimizing an approximate mapping function $\widehat{f}$. This function takes the initial conditions I and grid information N as input, with the true output being the flow field variables F and the model’s output being $\widehat{F}$. The training objective of the model is to make the approximate mapping function $\widehat{F}=\widehat{f}(I,N,\lambda )$ as close as possible to the true mapping function $F=f\left(I,\mathbb{N}\right)$. This is a deep learning method for mining physical mapping rules from flow field data, where $\lambda$ are the hyperparameters of the model training. The deep learning training process in this paper can be expressed as

$$\underset{\lambda }{argmin}{∥f(I,N)-\widehat{f}(I,N,\lambda )∥}_{L^2}$$

(7)

2.2. Data Preprocessing

To reduce the errors faced by the PINN-AeroFlow-U model when predicting complex flow field regions such as the wall boundary layer and the blade leading edge, the flow field is subjected to coordinate transformation before training. This maps the CFD physical space grid coordinates to a Cartesian coordinate system. In this paper, the transformation matrix is defined as

$$T=\left[\begin{array}{ccc}{\partial }_x\xi & {\partial }_y\xi & {\partial }_z\xi \\ {\partial }_x\eta & {\partial }_y\eta & {\partial }_z\eta \\ {\partial }_x\zeta & {\partial }_y\zeta & {\partial }_z\zeta \end{array}\right]={\left[\begin{array}{ccc}{\partial }_{\xi }x& {\partial }_{\eta }x& {\partial }_{\zeta }x\\ {\partial }_{\xi }y& {\partial }_{\eta }y& {\partial }_{\zeta }y\\ {\partial }_{\xi }z& {\partial }_{\eta }z& {\partial }_{\zeta }z\end{array}\right]}^{-1}$$

(8)

and normalized according to the index $(i,j,k)$ of the physical space coordinate system, yielding $\xi ={\displaystyle \frac{i-1}{\left(i_{\max }-1\right)}}$, $\eta ={\displaystyle \frac{j-1}{\left(j_{\max }-1\right)}}$, and $\zeta ={\displaystyle \frac{k-1}{\left(k_{\max }-1\right)}}$.

The mapping from equidistant Cartesian space to the curvilinear coordinate frame is achieved by a geometry-conditioned projection, wherein each grid point is represented by a normalized distance vector to the nearest blade surface along the local streamwise direction. This transformation embeds the geometric context into the input tensor while maintaining spatial locality. Although no explicit Jacobian matrix or determinant is computed, the network retains sensitivity to geometric deformation through learned spatial filters applied over the transformed domain. Since both the input and output of the neural network are defined in the same transformed space, an inverse mapping is not explicitly required.

Figure 4 shows the mapping relationship between the physical space and computational space coordinates of the CDA (Controlled Diffusion Airfoil). Figure 4a shows the physical space coordinates (i.e., under the $(x,y,z)$ coordinates), and Figure 4b shows the computational space (i.e., under the $(\xi ,\eta ,\zeta )$ coordinates, where $\xi ,\eta ,\zeta \in$ (0,1)).

Figure 4a shows a schematic diagram of generating a body-fitted grid in physical space, where ${\Gamma }_1$ represents the blade surface boundary coordinates, ${\Gamma }_3$ represents the coordinates of the flow field boundary, and ${\Gamma }_2$ and ${\Gamma }_4$ represent the internal interface of the computational domain. There is a one-to-one correspondence between the points in the physical plane and the points in the computational plane.

As shown in Figure 4b, the coordinate transformation method maps complex geometric regions (such as the flow region around a blade) to a Cartesian grid, allowing numerical calculations to be performed using equidistant grids without the need for dense unstructured grids around every complex geometric feature. In this process, local grid refinement is performed near the blade and in the boundary layer region through grid stretching and deformation. This improves the resolution of critical flow field regions while maintaining computational efficiency, thereby better capturing the flow details of complex flow features such as boundary layers, separation zones, and shock waves [28]. Considering that the coordinate transformation is bijective and can be accurately inversely mapped back to the original physical space, the solution obtained by the model in the computational space is equivalent to the solution in the original physical space.

To enhance numerical stability and geometric fidelity, additional implementation details were adopted in the coordinate transformation process. The shortest distance d from each grid point to the airfoil surface was computed using a nearest-neighbor projection algorithm. Specifically, the airfoil surface was discretized into a triangulated mesh, and for each point in the computational domain, the Euclidean distance to the closest triangle was computed using a k-d tree structure to accelerate nearest-point queries. This ensures efficient and accurate approximation of wall proximity, which is critical for capturing boundary layer effects. To address edge effects near the domain boundaries during the transformation, we introduced buffer zones in the computational grid. Nodes located within two grid layers of the outer boundary were excluded from gradient-based loss computations, preventing potential artifacts due to extrapolation or metric distortion. Furthermore, we applied a smoothing filter to the metric coefficients near the edges to suppress numerical noise and ensure a stable input field for network training.

Different compressor blade geometries exhibit the same boundary conditions in the computational plane $(\xi ,\eta ,\zeta )$, while the difference in governing equations is reflected in different metric coefficients within the governing equations. The model uses first-order metric coefficients to represent the grid information in the computational space $\left({\xi }_x,{\xi }_y,{\xi }_z,{\eta }_x,{\eta }_y,{\eta }_z,{\zeta }_x,{\zeta }_y,{\zeta }_z\right)$, the coordinates of the airfoil surface points to represent the airfoil geometric information $\left(x_0,y_0,z_0\right)$, and the shortest distance d from each coordinate point in the flow field to the airfoil surface as a supplement to the coordinate information [26]. Finally, the grid information and initial conditions can be expressed as

$$\mathrm{Metrics}=\left[{\xi }_x,{\xi }_y,{\xi }_z,{\eta }_x,{\eta }_y,{\eta }_z,{\zeta }_x,{\zeta }_y,{\zeta }_z\right]$$

(9)

$$N_{i_{max}\times j_{max}\times k_{max}\times 13}=\left[\mathrm{Metrics},x_0,y_0,z_0,d\right]$$

(10)

$$I_{i_{max}\times j_{max}\times k_{max}\times 5}=\left[M{a}_{\infty },Ao{A}_{\infty },C_{max},\theta ,\beta \right]$$

(11)

2.3. PINN-AeroFlow-U Network Architecture

This study proposes a flow field prediction model named PINN-AeroFlow-U, the architecture of which is shown in Figure 5. The model is constructed in an encoder–decoder manner, with a total of 4 encoders and 4 decoders. Each encoder consists of a double convolutional layer containing two convolutional operations and a max-pooling layer for downsampling, while each decoder consists of a double convolutional layer that also performs two convolutional operations, paired with a transposed convolutional layer for upsampling. Skip connections are used to concatenate the encoder and decoder feature maps at the same level to preserve original data information [9]. To further enhance the model’s ability to extract spatial information, PINN-AeroFlow-U appends a Convolutional Block Attention Module (CBAM) to the last upsampling block of the decoder.

The input of the model consists of 18 variables related to airfoil geometry and inflow conditions from Equations (10) and (11), including the grid information in the computational space $\left({\xi }_x,{\xi }_y,{\xi }_z,{\eta }_x,{\eta }_y,{\eta }_z,{\zeta }_x,{\zeta }_y,{\zeta }_z\right)$, the coordinates of the airfoil surface points $\left(x_0,y_0,z_0\right)$, the shortest distance d from each coordinate point in the flow field to the airfoil surface, the inlet Mach number $M{a}_{\infty }$, the angle of attack $Ao{A}_{\infty }$, the maximum relative thickness $C_{max}$, the airfoil camber angle $\theta$, and the inlet flow angle $\beta$. The output consists of the 5 flow field variables listed in Equation (5), including the pressure field, the temperature field, and the velocity fields in the x, y, and z directions.

PINN-AeroFlow-U employs independent decoders for different flow field variables (such as pressure, temperature, and the three velocity components). This architecture allows the model to be optimized for each channel, thereby improving the prediction accuracy for flow field data containing complex features. The separated decoder structure also helps to reduce interference between variables during the normalization process.

The skip connection mechanism of PINN-AeroFlow-U directly concatenates the feature maps from the encoder to the feature maps in the decoder. This is equivalent to fusing high-level global shape and structural information with low-level edge and texture information, which can capture multi-scale details while preserving the original details of the input data.

By introducing the CBAM module, the PINN-AeroFlow-U model enhances its ability to capture spatial and channel dependencies in the flow field data. The spatial attention module can learn the weights of different positions in the spatial dimension of the image, thereby highlighting the key regions in the image [30]. This mechanism can help the model more accurately locate important features in the flow field, such as boundary layers and separation regions. The channel attention module enhances the model’s ability to capture flow field features by learning the importance of different channels, which helps the model distinguish the important information of different physical quantities (such as velocity, pressure, and temperature). With the help of the attention mechanism, the model can adaptively filter and focus on the features that play a decisive role in the prediction task. This improves the model’s adaptability to new data, giving it good generalization performance.

The input to the PINN-AeroFlow-U model is a tensor of dimensions $128\times 64\times 32\times 18$. Here, 128 corresponds to the flow passage length, 64 to the flow passage width (or pitch), and 32 to the blade height. The last dimension, 18, represents the number of input variables or channels (e.g., coordinates, flow parameters).

After passing through the PINN-AeroFlow-U’s first input layer, this tensor is transformed into $128\times 64\times 32\times 64$. This is because the U-Net typically standardizes the channel dimension to 64 for subsequent processing stages. Subsequently, during each downsampling step, the spatial dimensions (length, width, height) are halved, while the number of channels is doubled.

2.4. Dataset Preparation

To verify the generalization ability and final prediction accuracy of the PINN-AeroFlow-U model, a batch of airfoils was constructed, and the flow field variable data of the compressor airfoils was obtained by CFD calculations under certain operating conditions. Data was extracted from the CFD data using a body-fitted grid method, a curvilinear coordinate system was constructed, and the coordinates were normalized to establish the dataset required for model training. Under supersonic and high angle-of-attack conditions, the flow field around the compressor blades is extremely complex. Shock wave discontinuities frequently occur, and flow separation phenomena are also relatively strong. Affected by this, the flow field data obtained by CFD calculations under these complex conditions has the problem of poor accuracy. Considering the calculation accuracy of CFD and the fitting ability of the PINN-AeroFlow-U model, the operating conditions and design parameters of the training dataset are shown in

Table 1. Initial condition parameter table of airfoil dataset.

Number of Airfoils	Initial Conditions
36 + 4	$M{a}_{\infty }\in (0.5,0.7)$
	$Ao{A}_{\infty }\in (-3.0^{\circ },6.0^{\circ })$
	${\theta }_{\infty }\in (4^{\circ },{30}^{\circ })$
	$C_{max}\in (0.03,0.1)$

.

In total, 2560 CFD simulations were generated by combining 40 unique airfoil geometries with 64 variations in freestream and geometric parameters. Of these, 2304 samples from 36 airfoils were used for training, while the remaining 256 samples from 4 entirely separate airfoils formed the test set. This split ensures that the evaluation reflects the model’s ability to generalize to previously unseen blade shapes and operating conditions.

The computational domain for each CFD case includes a single blade passage, with pitchwise periodic boundaries to reflect cascade behavior. The axial inlet boundary is positioned 1.0 chord length upstream of the leading edge, and the outlet is placed 1.5 chord lengths downstream of the trailing edge. These distances ensure that the inflow develops naturally before interacting with the blade and that downstream pressure recovery is captured without artificial reflection. The spanwise domain height equals one blade span and is discretized uniformly. This domain setup follows established practices in compressor CFD studies [31,32] to minimize boundary-induced distortions while maintaining computational efficiency.

A CDA (Controlled Diffusion Airfoil) is used for the compressor airfoil. This type of airfoil prevents boundary layer separation by controlling the diffusion of airflow on the suction surface and overcomes the airflow separation phenomenon at the blade tip and root [1]. Forty CDAs with different shapes were designed for CFD simulation, of which the flow field data corresponding to 36 airfoils was used for model training, and the remaining 4 airfoils were used to test the model’s generalization ability.

Ten operating conditions were set with uniform intervals within the ranges shown in Table 1, and CFD calculations were performed to generate flow field data. The CFD simulation results were extracted using a structured grid data structure, and a schematic diagram of the structured grid node distribution is shown in Figure 6. The distribution of grid nodes in the computational space after coordinate transformation is shown in Figure 7.

The dimensions of the computational space in the three directions are 128, 64, and 32, respectively. The parameter dimension at each grid node is 18, including physical variables such as ($Metrics,x_0,y_0,z_0,d,M{a}_{\infty },Ao{A}_{\infty },C_{max},\theta ,\beta$). The dimensions of the entire dataset are 40 × 10 × 128 × 64 × 32 × 18, and the specific data allocation is shown in

Table 2. Dataset dimension allocation.

Dataset	Dimension
Train	36 × 10 × 128 × 64 × 32 × 18
Test	4 × 10 × 128 × 64 × 32 × 18

.

Since there is a significant difference in the numerical ranges of different variables in Equations (10) and (11), to avoid negative effects on model training due to feature imbalance, leading to some features dominating the training process while others are ignored, it is necessary to standardize the input variables. The specific calculation formula is as follows:

$${\gamma }_j={\displaystyle \frac{{\gamma }_i-{\gamma }_{min}}{{\gamma }_{max}-{\gamma }_{min}}}$$

(12)

where ${\gamma }_{max}$ and ${\gamma }_{min}$ represent the global maximum and minimum values of the input variable, respectively.

To ensure the numerical accuracy of the CFD data used for model training and evaluation, a mesh independence study was performed. A representative compressor airfoil case (${\mathrm{Ma}}_{\infty }$ = 0.6), (${\mathrm{AoA}}_{\infty }$ = 2.0°, Cmax = 0.08, $\theta$ = 20°) was simulated using three structured mesh resolutions: coarse, medium, and fine. The characteristics of each grid, along with comparisons of key flow quantities, are summarized below (

Table 3. Summary of mesh resolution and flow quantity differences for mesh independence study.

Grid Level	Total Cells	Blade Surface Nodes	Minimum First-Layer Y+	Cp Max Difference (%)	Total Pressure Loss Diff (%)
Coarse	1.1 million	∼4000	∼1.0	—	—
Medium	2.3 million	∼8000	∼0.8	2.7% (vs. fine)	1.9% (vs. fine)
Fine	4.6 million	∼16,000	∼0.6	—	—

). The differences in surface pressure coefficient (${\mathrm{C}}_{\mathrm{p}}$) and total pressure loss between the medium and fine grids were found to be within 2.7% and 1.9%, respectively. These values indicate that the CFD results using the medium-resolution mesh are sufficiently accurate and mesh-independent for use in constructing the training and testing datasets. Therefore, the medium-resolution grid was adopted for all CFD simulations in this study.

The CFD simulations were performed using steady-state RANS equations with the Spalart–Allmaras turbulence model. Boundary conditions were defined as follows: a uniform total pressure and total temperature were imposed at the inlet, along with prescribed Mach number and angle of attack; a static pressure condition was applied at the outlet to control the operating point. Blade surfaces were modeled as no-slip, adiabatic walls with near-wall y⁺ values below 1.0 to ensure boundary layer resolution. Periodic boundaries were enforced on the pitchwise sides to replicate cascade flow behavior, while hub and shroud surfaces were treated as slip walls. These settings follow standard practices for subsonic axial compressor simulations and were applied consistently across all cases to ensure physical reliability and dataset uniformity.

3. Model Training and Results Discussion

This section focuses on a detailed exposition of the model training strategies and an in-depth analysis of the experimental results. Considering the performance differences of the AeroFlow-U model before and after the introduction of the PINN, we will meticulously discuss both the AeroFlow-U model (without the PINN loss function) and the model after incorporating PINN-AeroFlow-U. This includes the construction of the loss function, the selection of training hyperparameters, the configuration of the computing platform, the loss reduction trends, and a comparative analysis of the predicted results (pressure, temperature, and velocity components in different directions) with CFD.

3.1. Model Training Strategy

3.1.1. AeroFlow-U Loss Function

The training objective of the AeroFlow-U network is to minimize the loss function between the predicted output and the ground truth data. This loss function consists of three parts: Mean Square Error loss (MSE), Gradient loss (Grad), and Airfoil Surface loss (Layer). The specific formula is

$$\mathrm{Loss}=\mathrm{MSE}+{\alpha }_1·\mathrm{Grad}+{\alpha }_2·\mathrm{Layer}$$

(13)

The weighting coefficients ${\alpha }_1$ and ${\alpha }_2$ were selected through a grid search over candidate values ranging from ${10}^{-3}$ to 100. The optimal setting (${\alpha }_1=1.0$, ${\alpha }_2=0.01$) was chosen based on the minimum mean absolute error (MAE) averaged across all five predicted flow variables (pressure, temperature, and velocity components) on a held-out validation set. This configuration was found to yield a stable balance between data fidelity and physical consistency during training.

The MSE loss function is one of the most commonly used loss functions in regression problems. It measures the model’s prediction performance by calculating the squared difference between the model’s predicted value ${\widehat{\mathbf{Y}}}_{n,i,j,k}$ and the actual target value ${\mathbf{Y}}_{n,i,j,k}$.

$${Loss}_{MSE}={\displaystyle \frac{1}{n_y·H·W·D}}\sum _{n=1}^{n_y}\sum _{i=1}^H\sum _{j=1}^W\sum _{k=1}^D{\left({\mathbf{Y}}_{n,i,j,k}-{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2$$

(14)

Let H, W, and D denote the height, width, and depth of the 3D grid used to represent each flow field sample. In our setting, these correspond to the number of grid points along the Y, X, and Z directions, respectively. The indices i, j, k iterate over the grid points in these respective dimensions.

The Grad loss function is used to improve the continuity of the flow field predicted by the model. This loss function measures the smoothness of the model’s predictions by calculating the derivatives of the flow field variables along the i, j, and k directions, denoted as ${\displaystyle \frac{\partial }{\partial h}}$, ${\displaystyle \frac{\partial }{\partial w}}$, and ${\displaystyle \frac{\partial }{\partial d}}$.

When calculating the Grad loss, a second-order central difference scheme is used to approximate the first and second derivatives. The central difference scheme is a numerical differentiation method that uses the values of a function near a specific point to estimate the derivative at that point. The second-order central difference scheme is chosen for its high accuracy and low sensitivity to computational errors. The calculation formula for the first-order derivative using this method is

$${\displaystyle \frac{\partial Y_{n,i,j,k}}{\partial h}}\approx {\displaystyle \frac{Y_{n,i+1,j,k}-Y_{n,i-1,j,k}}{2\Delta h}}$$

(15)

For the second-order derivative, the calculation method is extended to

$${\displaystyle \frac{{\partial }^2{Y}_{n,i,j,k}}{\partial h^2}}\approx {\displaystyle \frac{Y_{n,i+1,j,k}-2Y_{n,i,j,k}+Y_{n,i-1,j,k}}{{(\Delta h)}^2}}$$

(16)

Similarly, the derivatives in the w and d directions can also be calculated. The Grad loss function comprehensively considers both the first and second derivatives, and its expression is

$$\begin{array}{cc}\hfill {Loss}_{Grad}=& {\displaystyle \frac{1}{n_y·H·W·D}}\sum _{n=1}^{n_y}\sum _{i=1}^H\sum _{j=1}^W\sum _{k=1}^D\left({\left({\displaystyle \frac{\partial }{\partial h}}{\mathbf{Y}}_{n,i,j,k}-{\displaystyle \frac{\partial }{\partial h}}{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2\right.\hfill \\ \hfill & +{\left({\displaystyle \frac{\partial }{\partial w}}{\mathbf{Y}}_{n,i,j,k}-{\displaystyle \frac{\partial }{\partial w}}{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2+{\left({\displaystyle \frac{\partial }{\partial d}}{\mathbf{Y}}_{n,i,j,k}-{\displaystyle \frac{\partial }{\partial d}}{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2\hfill \\ \hfill & +{\left({\displaystyle \frac{{\partial }^2}{\partial h^2}}{\mathbf{Y}}_{n,i,j,k}-{\displaystyle \frac{{\partial }^2}{\partial h^2}}{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2+{\left({\displaystyle \frac{{\partial }^2}{\partial w^2}}{\mathbf{Y}}_{n,i,j,k}-{\displaystyle \frac{{\partial }^2}{\partial w^2}}{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2\hfill \\ \hfill & \left.+{\left({\displaystyle \frac{{\partial }^2}{\partial d^2}}{\mathbf{Y}}_{n,i,j,k}-{\displaystyle \frac{{\partial }^2}{\partial d^2}}{\widehat{\mathbf{Y}}}_{n,i,j,k}\right)}^2\right)\hfill \end{array}$$

(17)

The Layer loss function focuses on the model’s prediction performance near the airfoil surface, which is crucial for calculating distributed and concentrated forces. The formula for the Layer loss is

$${Loss}_{Layer}={\displaystyle \frac{1}{n_y·H·D}}\sum _{n=1}^{n_y}\sum _{i=1}^H\sum _{k=1}^D{\left({\mathbf{Y}}_{n,i,1,k}-{\widehat{\mathbf{Y}}}_{n,i,1,k}\right)}^2$$

(18)

3.1.2. PINN-AeroFlow-U Loss Function

To make the model predictions more accurate and stable, the Euler equation constraint (Euler equation error) is added to the output of the PINN-AeroFlow-U model. The Euler equations are a set of partial differential equations in fluid dynamics that describe the motion of inviscid fluids, representing the fundamental physical laws of fluid motion. Physics-informed neural networks (PINNs) require less training data compared to purely data-driven deep learning methods [23,26] and are more consistent with physical laws. The loss function formula for PINN-AeroFlow-U is

$$\mathrm{Loss}=\mathrm{MSE}+{\alpha }_1·\mathrm{Grad}+{\alpha }_2·\mathrm{Layer}+{\alpha }_3·\mathrm{Euler}$$

(19)

In three-dimensional flow field problems, the fluid state vector U is defined as

$$U={[\rho ,\rho u,\rho v,\rho w,E]}^T$$

(20)

where $\rho$ is the fluid density, $u,v,w$ are the velocity components of the fluid in three-dimensional space, and E is the total energy of the fluid. The fluid flux functions $F\left(U\right)$, $G\left(U\right)$, and $H\left(U\right)$ are defined as

$$\left\{\begin{array}{c}F\left(U\right)={\left[\rho u,\rho u^2+p,\rho uv,\rho uw,u(E+p)\right]}^T\hfill \\ G\left(U\right)={\left[\rho v,\rho uv,\rho v^2+p,\rho vw,v(E+p)\right]}^T\hfill \\ H\left(U\right)={\left[\rho w,\rho uw,\rho vw,\rho w^2+p,w(E+p)\right]}^T\hfill \end{array}\right.$$

(21)

where p is the fluid pressure. The conservation form of the Euler equations is

$${\displaystyle \frac{\partial U}{\partial t}}+\nabla ·F\left(U\right)+\nabla ·G\left(U\right)+\nabla ·H\left(U\right)=0$$

(22)

In the model of this paper, since the research focuses on steady flow problems, the time derivative term ${\displaystyle \frac{\partial U}{\partial t}}$ is omitted, and the Euler equations are simplified to

$$\nabla ·F\left(U\right)+\nabla ·G\left(U\right)+\nabla ·H\left(U\right)=0$$

(23)

The Euler loss function is constructed by calculating the residuals of the flow field variables predicted by the model with respect to the Euler equations. Specifically, the Euler loss function is defined as

$$\begin{array}{cc}\hfill {Loss}_{Euler}=& {\displaystyle \frac{1}{T_f}}\sum _{x\in T_f}\left|{\displaystyle \frac{F\left(U_{NN(i+1)}\right)-F\left(U_{NN(i-1)}\right)}{2\Delta x}}\right.+{\displaystyle \frac{G\left(U_{NN(j+1)}\right)-G\left(U_{NN(j-1)}\right)}{2\Delta y}}\hfill \\ \hfill & {\left.+{\displaystyle \frac{H\left(U_{NN(k+1)}\right)-H\left(U_{NN(k-1)}\right)}{2\Delta z}}\right|}^2\hfill \end{array}$$

(24)

where $T_f$ is the fluid domain, and $U_{NN(i+1)},U_{NN(i-1)},U_{NN(j+1)},U_{NN(j-1)},U_{NN(k+1)}$, and $U_{NN(k-1)}$ are the predicted values of the model at adjacent grid points. By minimizing the Euler loss, the model is guided to predict flow fields that satisfy the laws of physical conservation.

3.1.3. Computational Platform Configuration

In this study, the model was trained with parallel acceleration on a Tesla V100 platform. During the training process, the batch size was set to 10, the initial learning rate was 0.001, and the Adam optimization algorithm was used to update the model’s weights and biases. The number of training epochs was set to 1000, and a dynamic learning rate adjustment strategy was adopted: when training reached 800 epochs, the learning rate was multiplied by 0.1 to promote the model to reach the optimal convergence state. In terms of total training time, the full PINN-AeroFlow-U model was trained for 1000 epochs using four NVIDIA Tesla V100 GPUs in parallel. The complete training process on the full dataset (36 airfoils × 10 operating conditions) required approximately 42 h. To enhance training stability, gradient clipping and dynamic learning rate scheduling were applied during the final 200 epochs. Regarding scalability, the model demonstrates efficient generalization. For new compressor blade geometries not present in the training set, the model can be fine-tuned using only two to four new geometries, with convergence typically achieved within 2–4 h. This makes it feasible for rapid iteration in real-world aerodynamic design tasks.

The detailed software and hardware configuration information involved in model training is listed in

Table 4. Training hardware and software configuration of model.

Name	Device Parameters
Platform	WindowsServer
CPU	AMD EPYC 7K83 64-Core Processor 2.54 GHz
GPU	Tesla V100
Cuda	11.7
PyTorch	1.13.1 + cu117

.

3.1.4. Model Convergence

Figure 8 presents the convergence curves of the loss function during training. As shown in Figure 8, the loss of the PINN-AeroFlow-U model on both the training and test sets decreases rapidly before reaching 200 training epochs. After 200 training epochs, the loss function curves show some fluctuations. However, after completing 800 training epochs, with the dynamic adjustment of the learning rate, the loss function curves tend to stabilize. It was calculated that the average mean square error of a single batch of the model on the training set is $6.35\times {10}^{-6}$, and the average mean square error on the test set is $5.24\times {10}^{-4}$. The above results indicate that the model has a good convergence effect.

4. Results

4.1. Analysis

The study compared the differences between the CFD calculation results and the PINN-AeroFlow-U prediction results to analyze the prediction capability and limitations of PINN-AeroFlow-U.

This paper tested the calculation and prediction results of the compressor airfoil flow field under the following inflow conditions: Ma = 0.60, AoA = −2.0°, and design parameters ${\theta }_{\infty }$ = 16.675°, $C_{max}=0.062$. The experimental results analysis is shown in Figure 9. The leftmost column of contour maps shows the relevant variable results predicted by the AeroFlow-U network from top to bottom; the middle column represents the temperature, pressure, and three velocity component flow fields obtained by CFD calculations; and the rightmost column shows the relative error between the CFD calculation and the PINN-AeroFlow-U network prediction results.

The experimental results show that the average inference time of the PINN-AeroFlow-U model for the flow field under each operating condition of the test set is 1 s, while the average prediction time of the CFD method is 30 min. The inference speed of the PINN-AeroFlow-U model is significantly higher than the CFD simulation speed.

Observing the results in Figure 9, it can be found that the temperature, pressure, and x, y, z direction velocity contour maps obtained by the PINN-AeroFlow-U network prediction are highly consistent with the CFD simulation results. This indicates that the model can accurately capture the basic characteristics of the high-temperature, high-pressure, and low-speed regions formed at the blade leading edge due to airflow stagnation. This not only demonstrates the ability of the PINN-AeroFlow-U network to capture flow field details but also further confirms the accuracy of its three-dimensional predictions.

In-depth analysis of the relative error contour maps shows that the relative temperature error is controlled within a small range of 0 to 0.007, the relative pressure error is between 0 and 0.015, while the relative errors of the x, y, and z direction velocities are relatively larger, ranging from 0 to 0.02.

To evaluate the impact of the gradient constraint loss function on the computational results, we employed an ablation study method, comparing three scenarios: without gradient constraint, with gradient constraint, and based on original CFD data. By observing the spatial distribution and numerical variations of various physical quantities (such as pressure, temperature, and velocity) under different schemes, we can not only verify the role of gradient constraints in improving model performance and numerical stability but also clearly identify the importance and limitations of this constraint within specific computational regions.

For this purpose, we selected two representative airfoil designs (as shown in Figure 10), and we present the flow field distribution at three crucial spanwise sections (relative positions of 0.1, 0.5, and 0.9). Choosing these three spanwise locations allows us to reflect the subtle differences in flow field variations along the span. The results are visualized in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16.

To evaluate the effectiveness of the gradient constraint, we compared the prediction results of the PINN-AeroFlow-U model with and without the gradient constraint. The initial conditions for the inlet Mach number varying along the blade height h according to the formula are as follows: $Ma=-62.371·h^4+62.914·h^3-22.813·h^2+3.5373·h+0.4891$.

To avoid excessive image space due to displaying too many variables, the prediction results only showcase the comparison of the three velocity components, as the prediction of velocity variables is often the most challenging. This comparison is presented in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. The analysis reveals that the model without the gradient constraint performs relatively poorly. The prediction of the x-direction velocity field exhibits not only very obvious sawtooth-like artifacts but also significant fluctuations, severely impacting the prediction accuracy.

In contrast, the velocity field prediction with the gradient constraint not only aligns with the CFD results in terms of flow direction but also demonstrates higher accuracy and better detail in regions with large velocity changes, such as the wake and boundary layer, while eliminating those severe sawtooth-like fluctuations. It is particularly important to note that under these inlet conditions, shock waves may appear in the flow field. While both models might predict the shock waves, the prediction from the unconstrained model tends to be somewhat blurry. The constrained model, however, typically yields results for complex flow features like shock waves and potential separations that are closer to CFD in terms of position, strength, and clarity, appearing more realistic.

In conclusion, the comparison results clearly demonstrate that adding a gradient constraint to the PINN-AeroFlow-U model significantly improves the prediction accuracy and realism under the given inlet conditions. Compared to the model without the constraint, the constrained model not only predicts the overall flow field more accurately but also shows greater precision in displaying details (especially in regions of rapid change) and predicting complex phenomena such as shock waves, separations, and wakes. Consequently, its prediction results are closer to the CFD simulations.

To facilitate interpretation of the comparative prediction results shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, we provide in

Table 5. Case matrix for test scenarios presented in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16.

Case ID	Blade Profile	Ma∞	AoA∞ (°)	Cmax	$\mathit{\theta }$ (°)
A-1	Profile A	0.6	−2	0.062	16.675
A-2	Profile A	0.6	2	0.062	16.675
A-3	Profile A	0.6	4	0.062	16.675
B-1	Profile B	0.65	−1.5	0.07	25
B-2	Profile B	0.65	1.5	0.07	25
B-3	Profile B	0.65	4	0.07	25

the full parameter matrix corresponding to each test case. Each row represents a specific combination of inflow and geometric parameters used for model evaluation and visualization. Each profile corresponds to a specific CDA blade shape selected from the test dataset, and the conditions cover typical subsonic compressor operating regimes. These cases were selected to assess the model’s performance across varying degrees of flow separation, wake formation, and shock presence.

4.2. AeroFlow-U vs. PINN-AeroFlow-U

Although the coordinate transformation-based AeroFlow-U model has demonstrated good prediction performance, there is still room for improvement in regions with drastic flow changes and large gradients. To further enhance the model’s ability to handle complex physical phenomena, we introduced PINN constraints based on the Euler equation loss, see function (24), to further improve the model’s prediction performance in specific regions.

To intuitively compare the prediction accuracy of the AeroFlow-U and PINN-AeroFlow-U models, Figure 17 shows a comparison of the prediction results of the AeroFlow-U and PINN-AeroFlow-U models and the CFD numerical simulation results under a certain operating condition. As can be seen from Figure 17, the jagged traces in the flow field prediction are reduced, the boundaries of numerical changes become smoother and closer to the results calculated by CFD data, and the performance in the more complex flow field region near the hub is better than that of the AeroFlow-U model using only the U-Net network architecture. This phenomenon indicates that the introduction of PINN constraints helps to enhance the model’s overall grasp of flow field continuity.

In addition, the PINN-AeroFlow-U model also shows higher accuracy in predicting the high-temperature, high-pressure, and low-speed regions formed at the blade leading edge due to airflow stagnation. The numerical boundaries of these regions are closer to the CFD simulation results, indicating that the introduction of PINN improves the model’s ability to capture flow field details. This improvement is attributed to the PINN guiding the model’s training process by embedding the physical laws of the flow field, such as mass and momentum conservation.

The experimental results show that the coordinate transformation-based AeroFlow-U model has already demonstrated considerable accuracy in overall flow field prediction and has significant advantages in computational efficiency. Compared with traditional methods, its prediction error is significantly reduced, fully demonstrating the improvement effect of coordinate transformation on the model’s representation ability. After further incorporating PINN constraints, the prediction accuracy in regions with drastic flow changes has been further improved, especially reflected in the accurate capture of local physical quantities.

4.3. Physical Interpretation of Flow Features

To assess the physical reliability of the predicted flow fields, we compared PINN-AeroFlow-U outputs with CFD results in terms of key aerodynamic phenomena commonly observed in axial compressors, namely, shock formation, boundary layer separation, and wake development. These features are crucial to compressor performance and flow stability, and their accurate reproduction reflects the engineering utility of the model.

In cases involving higher inlet Mach numbers (e.g., $M{a}_{\infty }$ = 0.7), the model correctly identifies shock waves forming near the blade leading edge. This is evidenced by the sudden increase in predicted pressure and temperature, closely matching CFD distributions. These shock structures appear at physically expected positions, indicating that the model has effectively learned compressibility effects and flow deceleration phenomena.

In addition, the model demonstrates good capability in capturing near-wall flow behavior. Velocity field predictions reveal boundary layer separation on the suction surface in regions with adverse pressure gradients, particularly near the mid-span. The inclusion of gradient constraints enhances the accuracy and smoothness of these predictions. Furthermore, wake regions downstream of the blade trailing edge are clearly resolved, exhibiting the characteristic low-speed streaks and velocity deficits typical of viscous wake development. These results underscore the model’s ability to maintain physical consistency in complex flow regimes, thereby supporting its use in design and analysis workflows where physical interpretability is essential.

4.4. Additional Generalization Assessment

To assess the model’s generalization performance beyond the initially tested four airfoils, we conducted additional evaluations focusing on both boundary geometries and out-of-distribution (OOD) conditions. The four test geometries used previously were selected to cover extreme cases in the design parameter space, including airfoils with the largest camber angles ($\theta >{28}^{\circ }$), thinnest and thickest profiles ($\mathrm{Cmax}\in [0.03,0.1]$), and operating points near stall margins. Furthermore, we synthesized five additional OOD cases by extrapolating beyond the training range, such as (${\mathrm{AoA}}_{\infty }=7.5^{\circ }$) and $\mathrm{Cmax}=0.11$. While the prediction error increased moderately (pressure MAE rose from ∼1.1% to ∼2.4%), the model still produced physically plausible flow structures, capturing expected shock movements and boundary layer behavior. These results suggest that while the model performs best within the trained parameter envelope, it retains reasonable predictive capability in extrapolative settings, supporting its potential use in rapid design space exploration and early-stage concept evaluation.

4.5. Performance Comparison: With vs. Without PINN

To rigorously evaluate the contribution of the Euler equation residual in the loss function, we conducted a comparative analysis between two model variants: (1) AeroFlow-U, a baseline model utilizing the U-Net architecture with coordinate transformation and gradient/layer-based loss components; and (2) PINN-AeroFlow-U, which augments the baseline with a physics-informed constraint derived from the steady-state Euler equations. Both models were tested on an identical evaluation set, and their predictive accuracy was assessed across five fundamental flow field variables: pressure, temperature, and the three velocity components (u, v, w). The corresponding mean absolute error (MAE) and relative error (RE) metrics for each variable are reported in

Table 6. Quantitative comparison between models with and without Euler-based PINN constraint.

Variable	Model	MAE	Relative Error (%)
Pressure	AeroFlow-U	8.54 $\times {10}^{-3}$	1.86
	PINN-AeroFlow-U	4.87 $\times {10}^{-3}$	1.06
Temperature	AeroFlow-U	7.12 $\times {10}^{-3}$	1.34
	PINN-AeroFlow-U	4.15 $\times {10}^{-3}$	0.79
Velocity (u)	AeroFlow-U	1.62 $\times {10}^{-2}$	3.41
	PINN-AeroFlow-U	9.43 $\times {10}^{-3}$	2.01
Velocity (v)	AeroFlow-U	1.47 $\times {10}^{-2}$	3.12
	PINN-AeroFlow-U	8.76 $\times {10}^{-3}$	1.92
Velocity (w)	AeroFlow-U	1.58 $\times {10}^{-2}$	3.26
	PINN-AeroFlow-U	9.12 $\times {10}^{-3}$	1.95

. As shown, the PINN-AeroFlow-U model consistently outperforms the baseline across all variables, with notable improvements in high-gradient quantities such as velocity. The relative error reduction is particularly significant in velocity predictions (averaging over 1.3% improvement), reflecting the Euler constraint’s effectiveness in preserving conservation laws and improving flow continuity.

4.6. Limitations and Future Work

While the proposed PINN-AeroFlow-U framework demonstrates promising performance in predicting three-dimensional compressor flow fields, several limitations should be acknowledged:

(1) Dependence on CFD training data: The model’s accuracy is inherently tied to the fidelity of the CFD simulations used for training. Although mesh independence was verified, uncertainties in turbulence modeling or boundary conditions may propagate into the network predictions.

(2) Limited operating condition range: The training dataset covers a predefined range of inlet Mach numbers, angles of attack, and geometric parameters. While the model showed reasonable extrapolation behavior (see Section 4.4), its robustness under extreme or unseen conditions (e.g., high-speed transonic or stall) has not yet been fully validated.

(3) Assumption of steady, inviscid flow: The current model is based on the Euler equations and targets steady-state predictions. As such, it does not account for viscous effects, unsteady dynamics, or turbulence phenomena, which may limit applicability in certain engineering contexts.

(4) High memory demand for 3D data: Although the model runs efficiently at inference, training with high-resolution 3D data requires substantial GPU memory and computational resources, potentially limiting scalability for real-time applications or large-scale optimization loops.

Future work will address these limitations by extending the dataset to include viscous and unsteady cases, incorporating RANS-based physics constraints, and exploring model compression or multi-fidelity training strategies to improve scalability and adaptability to broader aerodynamic problems.

5. Conclusions

In this study, we proposed a physics-informed, data-driven framework for direct prediction of three-dimensional steady-state flow fields over compressor blade passages. By integrating a coordinate-transformed U-Net architecture with Euler-residual-based loss constraints, the model is able to learn full-field flow variable distributions from blade geometry and inlet conditions. Extensive evaluations on a CFD-generated dataset demonstrate that the proposed method achieves accurate predictions with relative errors below 3% across pressure, temperature, and velocity components. The PINN-enhanced variant outperforms baseline architectures by 10–15.

The framework also shows robust generalization to previously unseen airfoil geometries, highlighting its potential applicability to rapid aerodynamic design and optimization workflows. Future work will focus on extending the model to transonic and unsteady flow regimes, incorporating more detailed turbulence features, and exploring transfer learning strategies across broader compressor configurations.