For high-speed flow problems, the variation in air density cannot be ignored, so a multi-species compressible Navier-Stokes equation set [32] should be used, expressed in differential form as shown in Equation 1: R U = ∂ U ∂ t + ∇ ⋅ F ⃗ C U − ∇ ⋅ F ⃗ V U , ∇ U − Q = 0 (1) R ( U ) is the residual function, representing the remaining part of the equation, which should approach zero to satisfy the equation. U is the conservative variables of the flow field, which is shown in Equation 2: U = ρ 1 , … , ρ n , ρ u ⃗ , ρ e , ρ e v − e ⊺ (2) ρ 1 , … , ρ n represent the densities of different fluid components, ρ u ⃗ is the momentum, e is the energy, and t is time. ∇ is the Nabla operator, representing the gradient of a vector, which is used to describe the spatial rate of change of variables. F ⃗ C is the convective fluxs, which includes the momentum and energy of the fluid. u ⃗ is the velocity vector, including the three components u , v , w of the fluid in the Cartesian coordinate system.

The convective fluxes F ⃗ C , viscous fluxes F ⃗ V , and source terms Q are shown in Equation 3 [33]: F ⃗ C = ρ 1 u ⃗ ⋮ ρ n s u ⃗ ρ u ⃗ ⊗ u ⃗ + ρ I ̄ ρ u ⃗ h ρ u ⃗ e v − e , F ⃗ V = J ⃗ 1 ⋮ J ⃗ n s τ ̄ τ ̄ ⋅ u ⃗ + ∑ s J ⃗ s h s + q ⃗ v − e + q ⃗ t − r ∑ s J ⃗ s h s v − e + q ⃗ v − e , Q = ω ̇ 1 ⋮ ω ̇ n s 0 0 Ω ̇ (3) and p is the static pressure, e and e v − e are, respectively, the total energy per unit mass and the vibrational–electronic energy per unit mass for the mixture, h is the total enthalpy per unit mass, J ⃗ s is the species mass diffusion flux, τ ̄ is the viscous stress tensor, q ⃗ is the conduction heat flux, index s denotes the s -th chemical species, and n s is the total number of species.

The calculation uses the open-source software SU2 [34], the thermodynamic state of the continuous flow is modeled using the rigid rotor harmonic oscillator (RRHO) two-temperature model. Through the independence of energy levels, the total energy per unit volume is shown in Equation 4: ρ e = ∑ s ρ s e s t r + e s rot + e s vib + e s e l + e s ° + 1 2 u ⃗ ⊤ u ⃗ (4) Where ρ is the fluid density, e is the total energy per unit volume, ρ s is the density of the s -th species, e s t r is the translational energy of the s -th species, e s r o t is the rotational energy of the s -th species, e s v i b is the vibrational energy of the s -th species, e s e l is the electronic energy of the s -th species, and u ⃗ is the fluid velocity vector. The vibrational–electronic energy is shown in Equation 5: ρ e v − e = ∑ s ρ s e s vib + e s e l (5)

Generally, a gas mixture consists of polyatomic molecules, monatomic species, and free electrons. The expressions for translational, rotational, vibrational, and electronic energies are given below. First, for electrons, e s t r , e s r o t , and e s v i b are all zero. For monatomic species, e s r o t and e s v i b are zero. For all other cases, the expressions for each energy component are shown in Equation 6. e s t r = 3 2 R M s T (6a) e s rot = ξ 2 R M s T (6b) e s vib = R M s θ s vib exp θ s vib / T v e − 1 (6c) e s e l = R M s ∑ i = 1 ∞ g i , s θ i , s e l ⁡ exp − θ i , s e l / T v e ∑ i = 0 ∞ g i , s e x p − θ i , s e l / T v e for polyatomic and monatomic species, 3 2 R M s T v e for electrons. (6d)

Where R is the gas constant, T is the gas temperature, M s is the molar mass of the s -th species, ξ is the number of rotational axes (which should be an integer), θ s vib is the characteristic vibrational temperature of the species, θ i , s el is the characteristic electronic temperature of the corresponding species, and g i is the degeneracy of the i -th state.

In this work, a multilayer perceptron (MLP) [35] model is used for prediction. The multilayer perceptron is an artificial neural network structure composed of multiple layers of neurons. Structurally, the MLP consists of an input layer, hidden layers, and an output layer. The input layer receives the input data, with each data point corresponding to one neuron. There can be multiple hidden layers, and each neuron in a hidden layer receives the output from the neurons in the previous layer, then performs a nonlinear transformation through weighted summation and an activation function. The output layer provides the prediction result, with each neuron receiving the output from the final hidden layer and applying an activation function to it. The number of neurons in the output layer corresponds to the number of output values or categories in the prediction result.

In the hidden layers and output layer, each neuron typically applies an activation function to introduce non-linearity. In this work, the activation function used is the ReLU function [36]. The ReLU function maps negative values to 0 and keeps positive values unchanged. ReLU function is one of the most commonly used activation functions and is sufficient for the vast majority of machine learning models. Its mathematical form is shown in Equation 7 R e L U x = max 0 , x (7) When training the multilayer perceptron model, it is necessary to define a loss function to measure the difference between the model’s predictions and the true labels. In this work, the loss function used is the Mean Squared Error (MSE). MSE is used to calculate the average of the squared differences between the predicted values and the true values. Its mathematical expression is shown in Equation 8 M S E = 1 N ∑ i = 1 N y i − y ̂ i 2 (8) where y i is the true label, y ̂ i is the model’s predicted value, and N is the number of samples.

Convolutional Neural Networks (CNN) [37] can combine multiple convolutional layer and sampling layers to process input signals and achieve mapping to output targets in the fully connected layers. In a CNN, each feature map is a plane composed of multiple neurons, and input features are extracted using convolutional filters. Each convolutional layer contains multiple feature maps. The sampling layers perform subsampling based on the principle of local correlation, thereby reducing the amount of data while preserving valuable information.

The composite process of the convolutional and sampling layers in a CNN can be summarized as follows: For example, the first convolutional layer after the input layer has 8 feature maps, each of which is a 128 × 128 array of neurons. Each neuron extracts local features from the input layer using convolutional filters. In the sampling layer, each neuron is connected to a 2 × 2 neighborhood in the corresponding feature map of the previous layer. As a result, the sampling layer has 8 feature maps, each sized 64 × 64 . The next convolutional layer then applies convolution to the sampling layer, and the subsequent sampling layer continues to subsample the previous convolutional layer. Ultimately, the input is mapped to a multidimensional feature vector, which is then processed by the fully connected layer and the output layer to complete the recognition task.

In this work, the activation function used in the convolutional neural network is the ReLU function, and the loss function used is the Mean Squared Error (MSE).

The MLP excels at handling nonlinear relationships and is well-suited for predicting physical quantities on curves and reconstructing entire flow fields. In contrast, CNN efficiently process multidimensional information and appear more effective for reconstructing complete flow fields. While some researchers have compared these two approaches [38, 39] in domains like image processing, this study adopts both MLP and CNN methods to predict aerothermal heating and conducts a comparative analysis of their performance in flow field prediction.

In grid studies related to aerothermal heating, most researchers focus on the grid Reynolds number [40], which is shown in Equation 9 R e cell  = ρ ∞ V ∞ Δ n μ ∞ (9) That is, the grid Reynolds number is defined using Δ n as the characteristic length scale, where Δ n is typically taken as the height of the first layer of the grid in the boundary layer. The grid Reynolds number is used to reflect the density of the grid near the wall. For heat flux calculations, the finer the grid near the wall, the closer the calculated results are to the true values.

In this work, a two-dimensional cylinder with a radius of 0.045 m is selected as a case study, and experimental data from the Göttingen (HEG) high-enthalpy shock tunnel [41] is used to validate the accuracy of the numerical algorithm. The thermochemical nonequilibrium relaxation process occurring within the shock layer, which affects the density distribution of air components, is typically used to validate the physical and chemical models implemented in CFD codes. The two-dimensional cylinder is chosen as a common computational example for hypersonic flows due to the availability of extensive related data, which facilitates the validation of computational results. Additionally, the relatively low complexity of the two-dimensional cylinder flow field makes the computation more convenient, aiding in the subsequent establishment of datasets.

The mesh consists of 29,651 cells, and the Reynolds number is 10,643. The air model used is a five-species gas model with chemically non-equilibrium flow, neglecting high-temperature ionization effects and wall catalysis effects. The wall temperature is set to 300 K, and the numerical scheme employed is the AUSM+ format. The computational mesh is shown in the Figure 2, and the computational conditions are listed in Table 1.

The convergence of pressure and heat flux with respect to the number of iterations and grid density is shown in Figure 1. According to the graph, the pressure initially increases gradually and then stabilizes, reaching near stability around 500,000 iterations, with a relatively fast convergence rate. The heat flux starts with a large value, then decreases rapidly. After about 400,000 iterations, the rate of decrease slows down, and it essentially converges around 750,000 iterations.

The reference length of the two-dimensional cylinder in this work is 0.045 m . Regarding the effect of grid density on the pressure calculation results, starting with a 100 × 75 grid (Grid I),with a first-layer grid height of 2 × 1 0 − 7 m , the number of grid nodes was increased to 150 × 113 (Grid II), with a first-layer grid height of 1.5 × 1 0 − 7 m , 200 × 150 (Grid III), with a first-layer grid height of 1 × 1 0 − 7 m , and 300 × 225 (Grid IV), with a first-layer grid height of 6.7 × 1 0 − 8 m .with corresponding grid Reynolds numbers. From the graph, it can be observed that for the heat flux calculation, the grid density should be greater than 200 × 150 , and the corresponding grid Reynolds number should be less than 0.02365, for the pressure calculation, the computational results for these four grid types show good agreement and are all acceptable, the differences at theta of 90°, the discrepancy observed at θ = 90 ° arises from subtle influences of the outlet boundary conditions on the simulation results.

In conclusion, when the grid has 200 × 150 nodes, the grid Reynolds number is smaller than 0.02365, and the number of iterations exceeds 750,000 steps, the calculation results can be considered converged.

The computational results are compared with experimental and numerical results from other researchers [42], as shown in Figure 2. For the pressure calculations, the results align well with both experimental data and simulation data from the literature [42]. For the heat flux calculations, the predicted values are slightly lower than the experimental data, particularly in the stagnation region. This under-prediction is due to the lack of catalytic effects. Knight et al. [41] suggest that accurate prediction of surface heat flux requires consideration of catalytic effects. Compared to other non-catalytic results from Nompelis [43] and Maier [42], Nompelis employed a modified Steger–Warming flux splitting scheme, while Maier employed the AUSM+ format, the results in this work lie between the two and show a better match with experimental data, especially near the stagnation point. This confirms the correctness of the numerical simulation method used in this work.

For the aerodynamic heating prediction problem, during numerical calculations, as shown in Table 1, the temperature, pressure, velocity, density, and gas composition of the free-stream flow are the key factors affecting the calculation results. For the standard atmosphere, the temperature, pressure, density, and air composition are closely related to altitude, while the variation in density and air composition is minimal within small altitude ranges. In this work, the aerodynamic heating datasets were first established by varying the Mach number as a single variable, followed by changes in the incoming flow temperature and pressure, combined with the Mach number. Three variables are used to construct the aerodynamic heating datasets. Two aerothermal heating datasets were constructed in total. Apart from the changing variables, other parameters used for constructing the datasets are consistent with those in Tables 1. The mesh used consists of 150 nodes in the streamwise direction and 200 nodes along the wall direction, with the first layer of the boundary layer set to a height of 1 0 − 7 m . The dataset for the single Mach number variation spans from Mach 7.0 to Mach 11.9, with a step size of 0.1, resulting in a total of 50 computation results. The dataset constructed using three variables includes Mach numbers ranging from 8.5 to 9.5, with pressures of 460 Pa, 470 Pa, and 476 Pa, and temperatures of 890 K, 901 K, and 910 K, resulting in a total of 99 computation results.

Based on the datasets generated above, the study first uses MLP model to investigate single-parameter input and multi-parameter inputs. Then, MLP and CNN models were used for full-field prediction with single-parameter input. During the solving process, the pressure calculation converges more easily and is less affected by various factors. The network structure of the MLP models are shown in Tables 2, 3 and Figure 3. In Tables 2, 3, the 200 neurons in the output layer represent 200 discrete nodes distributed along the cylindrical wall surface. In Figure 3, the 30,000 neurons in the output layer correspond to 30,000 nodes spanning the entire computational flow field.

The schematic diagram of the CNN model is shown in Figure 4 and the network structure for CNN model predicting the full-field parameters is shown in Table 4. Pooling layers enhance computational efficiency and feature robustness through dimensionality reduction (their absence causes computational redundancy and sensitivity to minor input variations), while unpooling layers restore lost spatial details (missing them leads to blurry reconstructions or localization inaccuracies). The inputs consist of three channels, which represent the incoming Mach number and the x and y coordinates of the grid nodes. The output consists of two channels: pressure and temperature.

Using MLP model with Mach number as input to predict wall heat flux, the comparison between the predicted values and the true values when the Mach number is 7 is shown in Figure 5. The wall heat flux is predicted using Mach number, temperature, and pressure as inputs. When the Mach number is 8.5, the temperature is 901K, and the pressure is 476Pa, the comparison between the predicted value and the true value is shown in Figure 6. The “HeatFlux” is the wall heat flux, the X represents the distance from a point on the axis of symmetry of the semicircle to the center (with the direction opposite to the flow direction considered positive), and the raw data represents the simulation results. Due to the sharp discontinuity in shock position and the degraded predictive accuracy near the shock wave, significant oscillations are generated at the shock location. Using MLP with the Mach number as input to predict the temperature and pressure of the full field, the comparison between the predicted values and the true values when the Mach number is 7 is shown in Figure 7. Using CNN with the Mach number as input to predict the temperature and pressure of the full field, the comparison between the predicted values and the true values when the Mach number is 7 is shown in Figure 8. In Figures 7, 8, from left to right, the first column is the true values, the second column is the predicted values, and the third column is the relative difference between the predicted and true valuessimulation results. The relative difference is shown in Equation 10 Relative Error = y i − y ̂ i y i (10)

Using MLP model with Mach number as input, wall heat flux, the reverse line from the stagnation point along the flow direction temperature, and pressure are predicted, with an average relative error of 0.89%. Next, MLP mdoel is used with incoming Mach number, temperature, and pressure as inputs to predict wall heat flux, the reverse line from the stagnation point along the flow direction temperature, and pressure, achieving an average relative error of 0.73%. MLP model performs well in predicting physical quantities along the curve. Then, MLP model is used with Mach number as input to predict the temperature and pressure in the entire flow field. Computing aerothermal heating data for a single flow condition (e.g., freestream Mach 7, temperature 901 K, pressure 476 Pa) requires approximately 437.5 core-hours using CFD methods. In contrast, training the MLP model for full-field temperature/pressure prediction takes 2.22 h (2 s × 4000 epochs), while the CNN model requires a comparable 2.22 h (4 s × 2000 epochs). Once trained, both MLP and CNN models achieve rapid prediction of a single flow condition in a mere 0.15 s. The average relative error for pressure prediction is 7.6%, and for temperature prediction, it is 1.5%. Using CNN model with Mach number and flow field node coordinates as input, the average relative error for pressure prediction is 4.43%, and for temperature prediction, it is 3.34%. The comparisons of prediction errors between CNN and MLP models for temperature and pressure field predictions within the Mach number range of 7.7–8.6 are shown in Table 5. Both MLP and CNN models exhibit less accurate predictions for pressure fields compared to temperature fields across the entire flow domain. As shown in Figures 5, 6, the pressure undergoes abrupt changes (nearly two orders of magnitude difference) across the shock front, whereas the temperature gradually decreases post-shock, eventually aligning with the wall temperature at the boundary with less than one order of magnitude variation. This disparity in gradient magnitudes (Pressure exhibits an order-of-magnitude steeper gradient than temperature) leads to significantly poorer predictive performance for pressure in both models. For temperature field prediction, the CNN model exhibits nearly twice the error of the MLP model. Conversely, the MLP model demonstrates approximately double the error of the CNN model for pressure field predictions. As previously discussed, the challenge in pressure prediction stems from sharp gradients at shock positions, where the CNN’s localized feature extraction capability outperforms the MLP’s global approximation. Critically, total error of the CNN model is lower than that of the MLP, highlighting its balanced performance in multiphysics flow modeling. Overall, CNN model outperforms MLP model in predicting the flow field, and CNN model achieves higher accuracy in the post-shock and wall-adjacent regions. The error mainly occurs at the shock location.