RONAALP Algorithm

Description of the RONAALP module to create a reduced-order model of a high dimensional nonlinear functions with active learning procedure.

Mathematically, a RONAALP reduced-order model \(g\) predicts the outputs of the function of interest \(f\) such that \(\lVert g({\mathbf{z}}) - f(\mathbf{z}) \rVert_2\) is minimized and the computational cost is significantly reduced compared to the original function. The strategy employed to derive the reduced-order model is presented in [1], [2].

• Training
  • Encoder
  • Newman’s clustering algorithm
  • K-Means
  • Kernel functions
    • Exponential Kernel
    • Spline Kernel
• Evaluate
• Update

1

Scherding, C., Rigas, G., Sipp, D., Schmid, P. J., & Sayadi, T. (2023). Data-driven framework for input/output lookup tables reduction: Application to hypersonic flows in chemical nonequilibrium. Physical Review Fluids, 8(2), 023201.

2

Scherding, C. (2023). Predictive modeling of hypersonic flows in thermochemical nonequilibrium: physics-based and data-driven approaches. PhD Thesis, Sorbone University.

RONAALP.Ronaalp(d=2, feat_min=0, feat_max=1, method='Spline', n_rbf=200, n_ghost=1000, n_tree=20, rbf_degree=3, rbf_smoothing=0.001, n_epochs=300, architecture=[12, 6], clustering=True)

Reduced Order Nonlinear Approximation with Active Learning Procedure.

Parameters

dint, default = 2

Dimension of the latent space.

feat_minfloat, default = 0

Minimum value for MinMaxScaling.

feat_maxfloat, default = 1

Maximum value for MinMaxScaling.

n_treeint, default = 20

Number of trees in RandomForest classifier.

n_ghostint, default = 1000

Number of neighboring point to add to build the ghost layer of each cluster

n_rbfint, default = 200

Number of kernels in each clusters.

methodstr, optional

Kernel function to be used for RBF. Should be: - ‘Exponential’ : exponential kernel exp(-r^2/l) - ‘Spline’ : thin-plate spline kernel r^2*log(r)

rbf_degreeint, default = 3

Highest degree of added polynomial terms for RBF. Only used when method=’Spline’

rbf_smoothingfloat, default = 1e-3

Smoothing parameter for RBF.

architecturelist, default = [12,6]

A list specifying the number of neurons in each hidden layer for the encoding and decoding parts.

clusteringboolean, default = True

Specify if the user wants to perform Newman clustering in the latent space.

References

1

Scherding, C., Rigas, G., Sipp, D., Schmid, P. J., & Sayadi, T. (2023). Data-driven framework for input/output lookup tables reduction: Application to hypersonic flows in chemical nonequilibrium. Physical Review Fluids, 8(2), 023201.

2

Scherding, C. (2023). Predictive modeling of hypersonic flows in thermochemical nonequilibrium: physics-based and data-driven approaches. PhD Thesis, Sorbone University.

Attributes

scaler_xMinMaxScaler object

Scaler for input data.

scaler_yMinMaxScaler object

Scaler for latent space.

scaler_zMinMaxScaler object

Scaler for output data.

encoder: Keras neural network.

Deep encoder to project high-dimensional input data in low-dimensional latent space.

n_clusters: int

Number of clusters found by Newman algorithm in the latent space.

classifier: Scikit-learn random forest classifier

Classifier trained on Newman clusters.

kmeans_list: list of length n_clusters

List containing the kmeans objects of each cluster.

surrogate_list: list of length n_clusters

List containing the RBF surrogate models of each cluster.