ReviewOfMultiFidelityModels_ToyProblems

The repository, titled ReviewOfMultiFidelityModels_ToyProblems, serves as an invaluable supplement to the academic journal article "Review of Multifidelity Models." Designed as a comprehensive collection of Jupyter Notebooks, the repository aims to present a tangible and easily understandable demonstration of multifidelity models. These models are showcased through a series of Python-based toy problems, thus facilitating a deeper understanding of the article's theoretical concepts. The repository contributes to the broader scientific understanding of multifidelity approaches by providing hands-on, executable examples.

Multi-Fidelity Modeling Toy Problem 1: Additive and Multiplicative Corrections

This code provides a demonstration of multi-fidelity modeling techniques for refining low-fidelity (LF) models using high-fidelity (HF) data through additive and multiplicative corrections. The example uses one-dimensional analytic functions to illustrate these concepts and includes plotting for visualization.

Code Structure

The code is organized into two main classes: AddMultCorrPlotter and AddMultCorrModels.

AddMultCorrPlotter Class

This class is responsible for plotting LF, HF, and correction models.

Methods

• func_HF(x): High-Fidelity function definition.
• func_LF(x): Low-Fidelity function definition.
• plot_functions(): Plots LF and HF functions.
• x: X-axis values for plotting LF and HF functions.

AddMultCorrModels Class

This class handles the modeling and plotting of additive and multiplicative corrections.

Methods

• polynomial_fit(x, y, deg=2): Fits a polynomial model to data.
• func_add(x): Computes the additive correction function.
• func_mult(x): Computes the multiplicative correction function.
• plot_additive_correction(): Plots the additive correction.
• plot_multiplicative_correction(): Plots the multiplicative correction.
• plot_combined_corrections(): Plots combined corrections.

Usage

1. Instantiate the AddMultCorrPlotter class to define LF and HF functions and plot them.
2. Instantiate the AddMultCorrModels class, passing the AddMultCorrPlotter instance.
3. Use the methods in the AddMultCorrModels class to plot additive and multiplicative corrections as needed.

Example

if __name__ == "__main__":
    plotter = AddMultCorrPlotter()
    plotter.plot_functions()
    
    models = AddMultCorrModels(plotter)
    
    plt.figure(figsize=[5, 3])
    models.plot_additive_correction()
    
    plt.figure(figsize=[5, 3])
    models.plot_multiplicative_correction()
    
    plt.figure(figsize=[5, 3.5])
    models.plot_combined_corrections()
    
    plt.show()

Multi-Fidelity Modeling Toy Problem 2: Comprehensive Correction

Overview

The presented code is a Python implementation designed to visualize High-Fidelity (HF) and Low-Fidelity (LF) functions and to conduct sensitivity analysis. The code leverages the Matplotlib and NumPy libraries for plotting and numerical operations, respectively. It encapsulates functionality into three classes: FunctionVisualizer, ComprehensiveFunctionVisualizer, and SensitivityAnalysis.

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FunctionVisualizer Class

Attributes

• lb: Lower bound of the domain for function visualization.
• ub: Upper bound of the domain for function visualization.
• x: NumPy array containing equidistant points within [lb, ub] at which function evaluations are carried out.

Methods

• __init__(lb, ub): Initializes an object with specified lower and upper bounds for the domain.
• func_HF(x): Evaluates the High-Fidelity function (HF) at a given point ( x ).
• func_LF(x): Evaluates the Low-Fidelity function (LF) at a given point ( x ).
• visualize(): Generates a plot of both HF and LF functions within the specified domain.

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ComprehensiveFunctionVisualizer Class

Attributes

• x_sampHF: NumPy array representing the points at which the HF function is sampled.

Methods

• Inherits all methods from FunctionVisualizer.
• func_X(x): Transforms the input ( x ) into a design matrix ( X ) by considering additional features.
• func_coef(X, Y, W=1): Computes the model coefficients using weighted least squares.
• func_comprehensive(x): Constructs a comprehensive model by combining HF and LF information.
• visualize_comprehensive(): Generates a plot incorporating HF, LF, and the comprehensive model along with HF sampling points.

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SensitivityAnalysis Class

Attributes

• coef: Dictionary containing coefficients 'A', 'B', and 'C' for LF function adjustments.

Methods

• Inherits the relevant methods for HF and LF functions, comprehensive model, and weighted least squares from previous classes.
• plot_sensitivity(x, x_sampHF, coef_key, coef_val_range, plot_title): Generates sensitivity plots to analyze the impact of changing individual coefficients within specified ranges.

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Example Usage

Function Visualization

visualizer = FunctionVisualizer(0, 1)
visualizer.visualize()

Comprehensive Modeling

visualizer = ComprehensiveFunctionVisualizer(0, 1, np.array([0.1, 0.5, 0.9]))
visualizer.visualize_comprehensive()

Sensitivity Analysis

x = np.linspace(0, 1, 100)
x_sampHF = np.array([0.1, 0.5, 0.9])
sensitivity = SensitivityAnalysis()
sensitivity.plot_sensitivity(x, x_sampHF, 'A', np.linspace(-1.3, 2.7, 10), 'A_Sensitivity')

Multi-Fidelity Modeling Toy Problem 3: MultiFidelityModel Class Documentation

Overview

The MultiFidelityModel class encapsulates the functionality for generating a Multi-Fidelity Surrogate Model (MFSM). It aims to synergize high-fidelity model (HFM) and low-fidelity model (LFM) data to create a more accurate and computationally efficient model.

Dependencies

• matplotlib.pyplot: Used for plotting.
• numpy: For numerical operations.
• sklearn.svm.SVR: For Support Vector Regression.
• sklearn.ensemble.RandomForestRegressor: For Random Forest Regression.

Class Attributes

• x: Numpy array consisting of 100 points linearly spaced between 0 and 1.
• nLF: Integer, number of LFM samples.
• nHF: Integer, number of HFM samples.
• svr: Object of class SVR, used for the Support Vector Regression model.
• rf: Object of class RandomForestRegressor, used for the Random Forest model.

Methods

__init__(self, x, nLF, nHF)

Initializes class instance variables.

• Parameters:
  • nLF: Number of Low-Fidelity Model samples.
  • nHF: Number of High-Fidelity Model samples.

func_HF(x)

Static method representing the high-fidelity function.

• Parameters:
  • x: Input variable values (Numpy ndarray).
• Returns:
  • Output response values (Numpy ndarray).

func_LF(x)

Static method representing the low-fidelity function.

• Parameters:
  • x: Input variable values (Numpy ndarray).
• Returns:
  • Output response values (Numpy ndarray).

generate_data(self)

Generates random sampling points for LFM and HFM and calculates corresponding function values.

fit_SVR(self)

Fits the SVR model to approximate the discrepancy between HFM and LFM.

fit_RF(self)

Fits the Random Forest model to the low-fidelity data, thus building the Low-Fidelity Surrogate Model (LFSM).

y_MFSM_A(self, x)

Calculates the integrated surrogate model using Option A.

• Parameters:
  • x: Input variable values (Numpy ndarray).
• Returns:
  • Output response values (Numpy ndarray).

y_MFSM_B(self, x)

Calculates the integrated surrogate model using Option B.

• Parameters:
  • x: Input variable values (Numpy ndarray).
• Returns:
  • Output response values (Numpy ndarray).

plot(self)

Generates a plot to visualize the actual LFM, HFM, and MFSM estimates. The plot also displays the discrepancy between the models.

Example Usage

if __name__ == "__main__":
    model = MultiFidelityModel(x=1000, nLF=200, nHF=20)
    model.generate_data()
    model.fit_SVR()
    model.fit_RF()
    model.plot()

Multi-Fidelity Modeling Toy Problem 4: Multi-Fidelity Analysis Using Co-Kriging

Introduction

The MultiFidelityAnalysis class is an encapsulation for performing multi-fidelity analysis via Co-Kriging (CoKG). The class provides methods to fit a CoKG model, make predictions, and visualize the Low-Fidelity (LF) and High-Fidelity (HF) models alongside CoKG predictions.

Requirements

• Python 3.6 or higher
• NumPy
• Matplotlib
• OpenMDAO

Installation

pip install numpy matplotlib openmdao

Class Methods and Attributes

__init__(self, lb=0, ub=1)

Initializes the MultiFidelityAnalysis object.

Parameters:

• lb: Lower bound of the variable space (default is 0).
• ub: Upper bound of the variable space (default is 1).

func_HF(x)

High-Fidelity (HF) model as a function of x.

Returns:

• HF model response.

func_LF(x)

Low-Fidelity (LF) model as a function of x.

Returns:

• LF model response.

fit_coKG(self, Xe, Xc)

Fits the CoKG model with HF and LF data.

Parameters:

• Xe: Points where the HF model is evaluated.
• Xc: Points where the LF model is evaluated.

predict_coKG()

Predicts the CoKG model response over the generated points.

Returns:

• Tuple containing predicted response and standard deviation.

plot_models(self, fHF, fLF, Xe, Xc, f_pred)

Plots the LF, HF, and CoKG models with their sampling points.

Parameters:

• fHF, fLF: Responses from HF and LF models.
• Xe, Xc: Sampling points for HF and LF models.
• f_pred: CoKG predictions.

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Usage Example

MFA = MultiFidelityAnalysis()
Xe = np.array([[0.2], [0.4], [0.85]])
Xc = np.vstack((np.array([[0.1], [0.25], [0.3], [0.5], [0.6], [0.7], [0.8], [0.9]]), Xe))
MFA.fit_coKG(Xe, Xc)
f_pred = MFA.predict_coKG()
MFA.plot_models(fHF, fLF, Xe, Xc, f_pred)

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For further inquiries or contributions, feel free to reach out.

Author

M. Giselle Fernández-Giselle, fernandez48@llnl.gov

License

This project is licensed under the MIT License.

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This example initializes a MultiFidelityModel with 200 LFM samples and 20 HFM samples. Subsequently, it generates data, fits the SVR and RF models, and plots the outcomes.

Multi-Fidelity Modeling Toy Problem 5: Multi-Fidelity Forrester Function

This Python class encapsulates the functionalities related to the multi-fidelity Forrester function. It provides methods for function evaluation, plotting, and model training.

Class Definition

MultiFidelityForrester

A class for encapsulating the functionalities related to the multi-fidelity Forrester function.

Methods

__init__(self, lb=0, ub=1, num_points=100)

Initializes the class with default lower and upper bounds and the number of points.

• Parameters:
  • lb (float): Lower bound for function evaluation
  • ub (float): Upper bound for function evaluation
  • num_points (int): Number of linearly spaced points

func_HF(self, x)

Calculates the high-fidelity function ( f_{\text{HF}}(x) ).

• Parameters:
  • x (float): Input value
• Returns:
  • float: ( f_{\text{HF}}(x) ) value

func_LF(self, x)

Calculates the low-fidelity function ( f_{\text{LF}}(x) ).

• Parameters:
  • x (float): Input value
• Returns:
  • float: ( f_{\text{LF}}(x) ) value

plot_functions(self)

Plots both the high-fidelity and low-fidelity functions. Saves the plot as a high-resolution PNG file.

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Example Usage

# Instantiate the class and plot the functions
if __name__ == '__main__':
    mf_forrester = MultiFidelityForrester()
    mf_forrester.plot_functions()

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This documentation is intended to provide a clear and concise understanding of the MultiFidelityForrester class, its at