Gaussian processes

romtools provides lightweight Gaussian-process surrogate models for approximating quantities of interest from parameter samples. Gaussian processes can also be used as low-fidelity models inside higher-level workflows.

Examples

The example below demonstrates fitting a Gaussian process and evaluating its posterior mean and uncertainty at new parameter values.

import numpy as np

from romtools.rom.qoi_surrogates import (
    GaussianProcessKernel,
    GaussianProcessRegressorLite,
)

x_train = np.array([[0.0], [1.0], [2.0]])
y_train = np.array([0.0, 1.0, 0.0])
x_query = np.array([[0.5], [1.5]])

gp = GaussianProcessRegressorLite(
    kernel=GaussianProcessKernel(),
    noise_variance=1e-8,
)
gp.fit(x_train, y_train)
mean, std = gp.predict_mean_and_std(x_query)

POD for vector QoIs

For vector-valued QoIs, GaussianProcessQoiModel first mean-centers the training QoIs and computes a proper orthogonal decomposition (POD) using an SVD. The POD basis is truncated according to pod_energy_fraction and, optionally, max_pod_modes. A separate scalar Gaussian process is then trained for each retained POD coefficient, and predictions are reconstructed in the original QoI space.

from romtools.rom.qoi_surrogates import GaussianProcessQoiModel

surrogate = GaussianProcessQoiModel(
    parameters=training_parameters,
    qois=training_qois,
    parameter_names=["mu_1", "mu_2", "mu_3"],
    pod_energy_fraction=0.999999,
    max_pod_modes=20,
    normalize_parameters=True,
    normalize_targets=True,
)

For scalar QoIs, no POD reduction is performed and a single Gaussian process is trained directly on the QoI.

For examples where Gaussian-process surrogates are created automatically inside multifidelity inverse workflows, see Ensemble Kalman inversion and Variational inference.