ExponentialCovariance

Overview

A simple exponential covariance function can be constructed as

var element = document.getElementById("moose-equation-2d2c6fb8-a1ff-4750-b5b8-d1ba04988118");katex.render("k(x,x^\\prime) = \\sigma_f^2 \\, exp \\left(- r_\\ell(x,x^\\prime)^\\gamma \\right) + \\sigma_n^2 \\, \\delta_{x,x^\\prime},", element, {displayMode:true,throwOnError:false});

which is valid for $var element = document.getElementById("moose-equation-6af0aa44-656b-471d-bfd7-80cddcbae098");katex.render("0 < \\gamma \\leq 2", element, {displayMode:false,throwOnError:false});$ . $var element = document.getElementById("moose-equation-1ceb7543-8357-46a0-a741-871033a02d58");katex.render("r_\\ell(x,x^\\prime)", element, {displayMode:false,throwOnError:false});$ is a scaled distance based on the length factor $var element = document.getElementById("moose-equation-6c1249ea-a479-4e24-8813-2429ad044cc8");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$ , defined as

var element = document.getElementById("moose-equation-0cd18bc5-cc5e-4e28-968a-2d2964bb89f9");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});

When $var element = document.getElementById("moose-equation-6520f783-2194-473b-8874-4e6e5927b265");katex.render("\\gamma = 2", element, {displayMode:false,throwOnError:false});$ is equivalent to SquaredExponentialCovariance, save a factor of $var element = document.getElementById("moose-equation-11d577ca-53ef-493b-836b-f19537cbd37f");katex.render("2", element, {displayMode:false,throwOnError:false});$ (which can be absorbed into $var element = document.getElementById("moose-equation-2a90f738-aaa9-4be4-90b6-70b380fb009a");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$ ).

Hyperparameters

Table 1: Hyperparameters for Exponential Covariance Function

Variable	Domain	Description
$var element = document.getElementById("moose-equation-bcbcc9f1-7882-4ebd-aa93-fc59032e42b8");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-a839cc44-41cd-46e9-a2bb-d2e1d5fbfc20");katex.render("\\mathbb{R}_{> 0}^n", element, {displayMode:false,throwOnError:false});$	Length factors corresponding to input parameters*
$var element = document.getElementById("moose-equation-fa951ef9-2790-48c5-9fd6-3d395f4a51fb");katex.render("\\sigma_f", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-53949399-50ae-4c6e-bb77-58ea9a29d266");katex.render("\\mathbb{R}_{\\geq 0}", element, {displayMode:false,throwOnError:false});$	Signal variance*
$var element = document.getElementById("moose-equation-84c03ffb-1043-4bc5-a5eb-7a014728bb2e");katex.render("\\sigma_n", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-2593f196-0727-4bfc-9c7b-05374c8a3f49");katex.render("\\mathbb{R}_{\\geq 0}", element, {displayMode:false,throwOnError:false});$	Noise variance*
$var element = document.getElementById("moose-equation-20c06525-3a2c-405b-99df-e2a5291069c9");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-727fd87c-7443-4b97-9cdc-1f6d5db6a0fd");katex.render("(0,2]", element, {displayMode:false,throwOnError:false});$	Exponential factor

*See the Gaussian Process Trainer documentation for more in depth explanation of $var element = document.getElementById("moose-equation-4071e94b-5cb2-4cc2-9cea-5468a5576ff5");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-04f08ff7-0956-4aa2-a4a2-98c41be11abc");katex.render("\\sigma_f", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-29427719-d5fa-4f58-86c9-b670e48f77e3");katex.render("\\sigma_n", element, {displayMode:false,throwOnError:false});$ hyperparameters.

Example Input File Syntax

[Covariance]
  [covar]
    type = ExponentialCovariance
    gamma = 1 #Define the exponential factor
    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 = '0.551133 0.551133' #Select a length factor for each parameter (k and q)
  []
[]

warning

The detailed documentation of this object is only available when Moose is compiled with Libtorch. For instructions on how to compile Moose with Libtorch, visit the general installation webpage or click here.

(moose/modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_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 = MultiAppSamplerControl
    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 an 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 = GaussianProcessSurrogate
    trainer = GP_avg_trainer
  []
[]

[Covariance]
  [covar]
    type = ExponentialCovariance
    gamma = 1 #Define the exponential factor
    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 = '0.551133 0.551133' #Select a length factor for each parameter (k and q)
  []
[]

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

Overview
Hyperparameters
Example Input File Syntax