SquaredExponentialCovariance

Squared Exponential covariance function.

Overview

A widely used, general purpose isotropic covariance function is

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$var element = document.getElementById("moose-equation-ebfc003e-8a7d-456a-b1d8-ddc232514367");katex.render("r_\\ell(x,x^\\prime)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a scaled distance based on the length factor $var element = document.getElementById("moose-equation-b76d3811-71fc-41b3-9428-f159e82ba1eb");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , defined as

var element = document.getElementById("moose-equation-f5c1f507-5c0e-4a59-a0d7-fd5dc7d7f737");katex.render("r_\\ell(x,x^\\prime) = \\sqrt{ \\sum_n \\left( \\frac{x_i - x^\\prime_i}{\\ell_i} \\right)^2}.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Hyperparameters

Table 1: Hyperparameters for Squared Exponential Covariance Function

Variable	Domain	Description
$var element = document.getElementById("moose-equation-dc1bf964-d898-4a6f-8e17-f93f3badf6da");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	$var element = document.getElementById("moose-equation-af70cf77-aca1-42ac-ab4d-afe8bbdc47a8");katex.render("\\mathbb{R}_{>0}^n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	Length factors corresponding to input parameters*
$var element = document.getElementById("moose-equation-ad003b44-43f6-4032-bda7-745b7950bdb1");katex.render("\\sigma_f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	$var element = document.getElementById("moose-equation-8141d693-d602-42dd-9fb6-455436a475c6");katex.render("\\mathbb{R}_{\\geq 0}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	Signal variance*
$var element = document.getElementById("moose-equation-f7c09fbe-f51c-4fc3-bab8-937e13dfe3d5");katex.render("\\sigma_n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	$var element = document.getElementById("moose-equation-390b6ae8-4d2b-4099-853b-3e82f41f9bd2");katex.render("\\mathbb{R}_{\\geq 0}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	Noise variance*

*See the Gaussian Process Trainer documentation for more in depth explanation of $var element = document.getElementById("moose-equation-77a362c0-0f2b-4bd1-bae9-b444e60475e8");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-f05778ff-9a63-4dc3-adc8-1a75fb0eb5cf");katex.render("\\sigma_f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and $var element = document.getElementById("moose-equation-9cf77911-5d01-4360-aa07-52c7a8d69ab4");katex.render("\\sigma_n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ hyperparameters.

Example Input File Syntax

[Covariance]
  [covar]
    type = SquaredExponentialCovariance
    signal_variance = 1 #Use a signal variance of 1 in the kernel
    noise_variance = 1e-6 #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971' #Select a length factor for each parameter (k and q)
  []
[]

Input Parameters

length_factorLength Factor to use for Covariance Kernel
C++ Type:std::vector<double>
Controllable:No
Description:Length Factor to use for Covariance Kernel
noise_variance0Noise Variance ($\sigma_n^2$) to use for kernel calculation.
Default:0
C++ Type:double
Controllable:No
Description:Noise Variance ($\sigma_n^2$) to use for kernel calculation.
signal_varianceSignal Variance ($\sigma_f^2$) to use for kernel calculation.
C++ Type:double
Controllable:No
Description:Signal Variance ($\sigma_f^2$) to use for kernel calculation.

Required Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential.i)
(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_1D_tuned.i)
(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_2D.i)
(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential_tuned.i)
(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential_training.i)
(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_1D.i)
(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_2D_tuned.i)
(modules/stochastic_tools/examples/surrogates/gaussian_process/GP_normal_mc.i)

(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [test_sample]
    type = MonteCarlo
    num_rows = 100
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
    parallel_type = ROOT
  []
  [samp_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = test_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [train_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'GP_avg'
    execute_on = final
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'covar'             #Choose a squared exponential for the kernel
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
  []
[]

[Surrogates]
  [GP_avg]
    type = GaussianProcess
    trainer = GP_avg_trainer
  []
[]

[Covariance]
  [covar]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-6                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971'         #Select a length factor for each parameter (k and q)
  []
[]


[Outputs]
  [out]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [test_sample]
    type = MonteCarlo
    num_rows = 100
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
    parallel_type = ROOT
  []
  [samp_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = test_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [train_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'GP_avg'
    execute_on = final
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'covar'             #Choose a squared exponential for the kernel
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
  []
[]

[Surrogates]
  [GP_avg]
    type = GaussianProcess
    trainer = GP_avg_trainer
  []
[]

[Covariance]
  [covar]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-6                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971'         #Select a length factor for each parameter (k and q)
  []
[]


[Outputs]
  [out]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_1D_tuned.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
  [L_dist]
    type = Uniform
    lower_bound = 0.01
    upper_bound = 0.05
  []
  [Tinf_dist]
    type = Uniform
    lower_bound = 290
    upper_bound = 310
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 6
    distributions = 'q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [cart_sample]
    type = CartesianProduct
    linear_space_items = '9000 20 100'
    execute_on = initial
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'rbf'
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
    tao_options = '-tao_bncg_type kd'
    tune_parameters = ' signal_variance length_factor'
    tuning_min = ' 1e-9 1e-9'
    tuning_max = ' 1e16  1e16'
    tuning_algorithm = 'tao'
  []
[]

[Covariance]
  [rbf]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-3                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971'         #Select a length factor for each parameter (k and q)
  []
[]

[Surrogates]
  [gauss_process_avg]
    type = GaussianProcess
    trainer = 'GP_avg_trainer'
  []
[]

# # Computing statistics
[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'gauss_process_avg'
    execute_on = final
  []
[]

[Reporters]
  [cart_avg]
    type = EvaluateSurrogate
    model = gauss_process_avg
    sampler = cart_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [train_avg]
    type = EvaluateSurrogate
    model = gauss_process_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]


[Outputs]
  csv = true
  execute_on = FINAL
[]

(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_2D.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 50
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [cart_sample]
    type = CartesianProduct
    linear_space_items = '1 0.09 100
                          9000 20 100 '
    execute_on = initial
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
  []
  [train_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [cart_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = cart_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'rbf'
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
  []
[]


[Covariance]
  [rbf]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-6                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971'         #Select a length factor for each parameter (k and q)
  []
[]

[Surrogates]
  [GP_avg]
    type = GaussianProcess
    trainer = 'GP_avg_trainer'
  []
[]

[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'GP_avg'
    execute_on = final
  []
[]

[Outputs]
  [out]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential_tuned.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [test_sample]
    type = MonteCarlo
    num_rows = 100
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
    parallel_type = ROOT
  []
  [samp_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = test_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [train_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'GP_avg'
    execute_on = final
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'covar'             #Choose a squared exponential for the kernel
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
    tao_options = '-tao_bncg_type ssml_bfgs'
    tune_parameters = ' signal_variance length_factor'
    tuning_min = ' 1e-9 1e-9'
    tuning_max = ' 1e16  1e16'
    tuning_algorithm = 'tao'
    show_tao=true
  []
[]

[Surrogates]
  [GP_avg]
    type = GaussianProcess
    trainer = GP_avg_trainer
  []
[]

[Covariance]
  [covar]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-3                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971'         #Select a length factor for each parameter (k and q)
  []
[]


[Outputs]
  [out]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_squared_exponential_training.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
    parallel_type = ROOT
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'covar'             #Choose a squared exponential for the kernel
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
  []
[]

[Covariance]
  [covar]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-6                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971'         #Select a length factor for each parameter (k and q)
  []
[]

[Outputs]
  file_base = gauss_process_training
  [out]
    type = SurrogateTrainerOutput
    trainers = 'GP_avg_trainer'
    execute_on = FINAL
  []
[]

(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_1D.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
  [L_dist]
    type = Uniform
    lower_bound = 0.01
    upper_bound = 0.05
  []
  [Tinf_dist]
    type = Uniform
    lower_bound = 290
    upper_bound = 310
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 6
    distributions = 'q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [cart_sample]
    type = CartesianProduct
    linear_space_items = '9000 20 100'
    execute_on = initial
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
  []
  [cart_avg]
    type = EvaluateSurrogate
    model = gauss_process_avg
    sampler = cart_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [train_avg]
    type = EvaluateSurrogate
    model = gauss_process_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    sampler = train_sample
    response = results/data:avg:value
    covariance_function = 'rbf'
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
  []
[]

[Covariance]
  [rbf]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-3                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971'         #Select a length factor for each parameter (k and q)
  []
[]

[Surrogates]
  [gauss_process_avg]
    type = GaussianProcess
    trainer = 'GP_avg_trainer'
  []
[]

[Outputs]
  csv = true
  execute_on = FINAL
[]

(modules/stochastic_tools/examples/surrogates/gaussian_process/gaussian_process_uniform_2D_tuned.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 50
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [cart_sample]
    type = CartesianProduct
    linear_space_items = '1 0.09 100
                          9000 20 100 '
    execute_on = initial
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
  []
[]


[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'rbf'
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    tao_options = '-tao_bncg_type kd'
    sampler = train_sample
    response = results/data:avg:value
    tune_parameters = ' signal_variance length_factor'
    tuning_min = ' 1e-9 1e-9'
    tuning_max = ' 1e16  1e16'
    tuning_algorithm = 'tao'
  []
[]


[Covariance]
  [rbf]
    type=SquaredExponentialCovariance
    signal_variance = 1                       #Use a signal variance of 1 in the kernel
    noise_variance = 1e-3                     #A small amount of noise can help with numerical stability
    length_factor = '0.38971 0.38971'         #Select a length factor for each parameter (k and q)
  []
[]

[Surrogates]
  [GP_avg]
    type = GaussianProcess
    trainer = 'GP_avg_trainer'
  []
[]

[Reporters]
  [train_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [cart_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = cart_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'GP_avg'
    execute_on = final
  []
[]

[Outputs]
  [out]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/stochastic_tools/examples/surrogates/gaussian_process/GP_normal_mc.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 0
    upper_bound = 20
  []
  [q_dist]
    type = Uniform
    lower_bound = 7000
    upper_bound = 13000
  []
  [L_dist]
    type = Uniform
    lower_bound = 0.0
    upper_bound = 0.1
  []
  [Tinf_dist]
    type = Uniform
    lower_bound = 270
    upper_bound = 330
  []
[]

[Samplers]
  [sample]
    type = MonteCarlo
    num_rows = 500
    distributions = 'k_dist q_dist L_dist Tinf_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppCommandLineControl
    multi_app = sub
    sampler = sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value Mesh/xmax BCs/right/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
  []
[]

[Trainers]
  [GP_avg]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'rbf'
    standardize_params = 'true'               #Center and scale the training params
    standardize_data = 'true'                 #Center and scale the training data
    sampler = sample
    response = results/data:avg:value
    tao_options = '-tao_bncg_type gd'
    tune_parameters = ' signal_variance length_factor'
    tuning_min = ' 1e-9 1e-3'
    tuning_max = ' 100  100'
    tuning_algorithm = 'tao'
  []
[]

[Covariance]
  [rbf]
    type=SquaredExponentialCovariance
    noise_variance = 1e-3                     #A small amount of noise can help with numerical stability
    signal_variance = 1
    length_factor = '0.038971 0.038971 0.038971 0.038971' #Select a length factor for each parameter (k and q)
  []
[]

[Outputs]
  file_base = GP_training_normal
  [out]
    type = SurrogateTrainerOutput
    trainers = 'GP_avg'
    execute_on = FINAL
  []
[]

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