You're reading the documentation for a development version. For the latest stable documentation, please have a look at v5.1.X.

Gaussian Process Regression

The Gaussian process regression (GPR) model is a non-parametric, data-driven binding model that represents the equilibrium relation between pore-phase and solid-phase concentration by Gaussian process regression trained on tabulated data. The model predicts an equilibrium loading from training inputs in pore-phase concentration space and corresponding solid-phase loadings. :contentReference[oaicite:0]{index=0} :contentReference[oaicite:1]{index=1}

For each component \(i\) and bound state \(m\), an equilibrium loading

\[c^{s,\ast}_{i,m} = f_{i,m}(c^p)\]

is constructed from user-provided training data and trained kernel parameters. In the current implementation, the prediction is evaluated from pore-phase training inputs CP_VALS, solid-phase targets CS_VALS, and trained kernel hyperparameters TRAINED_PARAMS provided in the training_data scope. The input dimension is specified by NDIM and the kernel type by KERNEL. Supported kernels are MLP, RBF, RBF_Linear, and MLP_Linear. :contentReference[oaicite:2]{index=2}

The GPR predictor evaluates the equilibrium loading in the standard kernel form

\[c^{s,\ast}(c^p) = k(c^p, X)^\top \alpha,\]

where \(X\) denotes the training inputs, \(k(c^p, X)\) is the vector of kernel evaluations between the current pore-phase concentration and all training samples, and \(\alpha\) is obtained from the linear system

\[\alpha = \left(K(X,X) + \sigma_n^2 I\right)^{-1} y.\]

Here, \(K(X,X)\) is the kernel matrix assembled from the training inputs, \(y\) is the vector of solid-phase training values, and \(\sigma_n^2\) is the Gaussian noise variance added to the diagonal before Cholesky factorization. :contentReference[oaicite:3]{index=3}

Depending on the selected kernel, the covariance function is given by one of the following forms.

For the radial basis function kernel,

\[k_{\mathrm{RBF}}(x,x') = \sigma_{\mathrm{RBF}}^2 \exp\!\left( -\frac{\lVert x-x' \rVert^2}{2\,\ell_{\mathrm{RBF}}} \right).\]

For the linear kernel,

\[k_{\mathrm{Lin}}(x,x') = \sigma_{\mathrm{Lin}}^2 \, x^\top x'.\]

For the multilayer perceptron kernel,

\[k_{\mathrm{MLP}}(x,x') = \sigma_{\mathrm{MLP}}^2 \frac{2}{\pi} \arcsin\!\left( \frac{\sigma_w^2 x^\top x' + \sigma_b^2} {\sqrt{\sigma_w^2 x^\top x + \sigma_b^2 + 1} \sqrt{\sigma_w^2 x'^\top x' + \sigma_b^2 + 1}} \right).\]

In addition, the implementation supports additive mixed kernels

\[k_{\mathrm{RBF+Lin}}(x,x') = k_{\mathrm{RBF}}(x,x') + k_{\mathrm{Lin}}(x,x'),\]

\[k_{\mathrm{MLP+Lin}}(x,x') = k_{\mathrm{MLP}}(x,x') + k_{\mathrm{Lin}}(x,x').\]

These kernel definitions are used both for prediction and for evaluation of the Jacobian contribution. :contentReference[oaicite:4]{index=4} :contentReference[oaicite:5]{index=5}

An offset is computed once during configuration as the GPR prediction at zero input and is subtracted from subsequent predictions. Thus, the equilibrium loading used by the binding model is

\[c^{s,\ast}(c^p) = f(c^p) - f(0).\]

This shifts the model response such that the predicted loading is zero at the chosen reference point. :contentReference[oaicite:6]{index=6} :contentReference[oaicite:7]{index=7}

Kinetic form

The model is used in a kinetic linear-driving-force form. For each component \(i\) and bound state \(m\), the exchange term is based on the deviation of the current solid-phase loading \(c^s_{i,m}\) from the GPR-predicted equilibrium loading \(c^{s,\ast}_{i,m}\):

\[\frac{\partial c^s_{i,m}}{\partial t} = k^{\mathrm{kin}}_{i,m}\left(c^{s,\ast}_{i,m}(c^p) - c^s_{i,m}\right).\]

Equivalently, in residual form the implementation evaluates

\[r_{i,m} = k^{\mathrm{kin}}_{i,m}\left(c^s_{i,m} - c^{s,\ast}_{i,m}(c^p)\right).\]

Thus, the Gaussian process regression model provides the equilibrium target, while the kinetic constant \(k^{\mathrm{kin}}_{i,m}\) controls how fast the equilibrium state is approached. The kinetic parameter is configured through GPR_KKIN. :contentReference[oaicite:8]{index=8} :contentReference[oaicite:9]{index=9}

For more information on model parameters required to define in CADET file format, see Gaussian Process Regression.