Data-driven framework for input/output lookup tables reduction: Application to hypersonic flows in chemical nonequilibrium

Hypersonic flows are of great interest in a wide range of aerospace applications and are a critical component of many technological advances. Accurate simulations of these flows in thermodynamic (non)equilibrium (accounting for high temperature effects) rely on detailed thermochemical gas models. While accurately capturing the underlying aerothermochemistry, these models dramatically increase the cost of such calculations. In this paper, we present a model-agnostic machine-learning technique to extract a reduced thermochemical model of a gas mixture from a library. A first simulation gathers all relevant thermodynamic states and the corresponding gas properties via a given model. The states are embedded in a low-dimensional space and clustered to identify regions with different levels of thermochemical (non)equilibrium. Then, a surrogate surface from the reduced cluster space to the output space is generated using radial-basis-function networks. The method is validated and benchmarked on simulations of a hypersonic flat-plate boundary layer and shock-wave boundary layer interaction with finite-rate chemistry. The gas properties of the reactive air mixture are initially modeled using the open-source $\text{Mutation}++$ library. Substituting $\text{Mutation}++$ with the lightweight, machine-learned alternative improves the performance of the solver by up to 70% while maintaining overall accuracy in both cases.

Figure 1

CPU time of a benchmark simulation, run with different aerothermochemical models. Including nonequilibrium effects in the simulation causes a significant increase in computational time.

Figure 2

General schematic of the model training and coupling to replace any expensive input/output library.

Figure 3

Numerical range of selected $\text{Mutation}++$ outputs $\mathbf{Z}$ (vertical) with respect to the inputs $\mathbf{X}$ (horizontal).

Figure 4

Low-dimensional representation $Y\in {\mathbb{R}}^{N\times d}$ ($d=2$) of $X\in {\mathbb{R}}^{N\times D}$, colored by temperature $T$. Obtained with (a) PCA, (b) AE, (c) PLS, (d) IO-E.

Figure 5

Schematic of the input/output-encoder architecture proposed. An input sample $\mathbf{X}$ is passed through the encoder network to generate a low-dimensional representation $\mathbf{Y}$. $\mathbf{Y}$ is decoded back to predict the library output $\widehat{\mathbf{Z}}$. At each training step, the $L_2$ loss $\parallel \mathbf{Z}-\widehat{\mathbf{Z}}{\parallel }_2$ between the library and the network prediction is calculated and backpropagated until convergence of the encoder and decoder weights, $W_E$ and $W_D$, respectively, is reached.

Figure 6

Demonstration of the Newman algorithm. (a) Original distance matrix $\mathrm{\Delta }$ of data set $\mathbf{Y}$. (b) Restored communities from running Newman's algorithm on $A$, showing two distinct clusters.

Figure 7

Tesselation of the low-dimensional space after applying the k-means algorithm with $N_R=250$ on the concatenation of the input $\mathbf{X}\in {\mathbb{R}}^{N\times D}$ and output $\mathbf{Z}\in {\mathbb{R}}^{N\times D_Z}$ vectors. Black stars represent the cluster centroids' low-dimensional representation ${\mathbf{y}}_1^c,...,{\mathbf{y}}_{N_R}^c$.

Figure 8

Confusion matrix $C$ of the random forest classifier ($n_{\text{tree}}=20$), tested on unseen data points during training. $C_{ij}$ is the number of points belonging to cluster $c_j$ predicted to be in cluster $c_i$.

Figure 9

Profiles of (a) streamwise velocity, (b) temperature, (c) species mass fractions from left to right $N, NO, O, O_2$, and $N_2$, at ${\text{Re}}_x=2000$. (a), (b) Red line: Simulation in chemical nonequilibrium; black line: perfect gas assumption.

Figure 10

Wall distribution of (a) skin-friction coefficient, (b) pressure, and (c) species mass fractions (from top to bottom on left side: $N_2, O2, O, NO$, and $N$). (a), (b) Red line: Simulation in chemical nonequilibrium; black line: perfect gas assumption.

Figure 11

Reconstruction loss $\parallel \widehat{\mathbf{Z}}{-\mathbf{Z}\parallel }_2$ of the embedding done by IO-E with respect to the number of latent space dimensions $d$.

Figure 12

Number of clusters $N_c$ obtained with Newman's algorithm as a function of the threshold $\epsilon$, given as multiples of the distance matrix mean $\overline{\mathrm{\Delta }}$.

Figure 13

Training points $\mathbf{Y}$ colored by their cluster number. (a) In the latent space found by IO-E. (b) At the Cartesian location they were sampled from, with contours of temperature $T$ in Kelvin.

Figure 14

Left axis: Reconstruction error of surrogate model $\parallel \widehat{\mathbf{Z}}{-\mathbf{Z}\parallel }_2$ on a testing set (dashed line). Right axis: Mean of squared RBF coefficients $\overline{{\mathrm{\Lambda }}^2}$ (dotted line). Both with respect to the number of RBF centers $N_R$ used in training.

Figure 15

Comparison of the relative temperature error $|\widehat{T}-T|/T$ in percent with contours of temperature. (a) Model 1: Full IO-E. (b) Model 2: $d=6,N_c=1,N_R=250$. (c) Model 3: $d=2,N_c=1,N_R=250$. (d) Model 4: $d=6,N_c=2,N_R=250$.

Figure 16

Comparison of profiles of (a) streamwise velocity, (b) temperature, (c) species mass fractions from left to to right $N, NO, O, O_2$, and $N_2$ at ${\text{Re}}_x=2000$. Solid line and symbols correspond to the solution obtained using $\text{Mutation}++$ and the data-driven model, respectively.

Figure 17

Comparison of the time complexity of $\text{Mutation}++$ with Stefan-Maxwell diffusion (dotted black line), the purely local version (solid red line), and the data-driven model (dash dotted blue line). The best nonlinear least-squares fit of the form $C{N}^{\alpha }$ is added.

Figure 18

Training points $\mathbf{Y}$ colored by their cluster number. (a) In the latent space found by IO-E. (b) At the Cartesian location they were sampled from, with contours of magnitude of the density gradient $\parallel \mathbf{\nabla }\rho \parallel$.

Figure 19

Comparison of wall distribution of (a) skin-friction coefficient, (b) pressure, and (c) species mass fractions (from top to bottom on left side: $N_2, O2, O, NO$, and $N$). Solid line and symbols correspond to the solution obtained using $\text{Mutation}++$ and the data-driven model, respectively.