Physics-informed quantum kernel via ODE-residual fusion
Source:R/piml_quantum_bridge.R
piml_quantum_kernel.RdBuilds a physics-informed Gram matrix by combining the Pilar 6 ZZFeatureMap kernel over raw features with an RBF similarity over the depth-profile residuals of a fitted Pilar 2 ODE:
Usage
piml_quantum_kernel(
X,
y,
depths,
ode_fit,
alpha = 0.7,
sigma = NULL,
reps = 2L,
backend = c("rcpp", "r")
)Arguments
- X
Numeric matrix (rows = samples, columns = features to encode through the ZZFeatureMap). Features should already be in
[0, pi]– usequantum_scale()if needed.- y
Numeric response vector (one per row of
X) used to compute residuals against the ODE fit.- depths
Numeric vector of depths (same length as
y) at which observations were taken.- ode_fit
A fitted object from
piml_profile_fit()orpiml_profile_fit_bayesian()providing apredict()method.- alpha
Numeric in
[0, 1]; mixing weight. Default0.7(quantum-heavy, physics-informed but not physics-dominated).- sigma
Numeric; RBF bandwidth on the residual scale. Default: median absolute residual over the training set (Silverman's rule of thumb).
- reps
Integer; ZZFeatureMap repetitions (forwarded to
quantum_kernel()).- backend
Forwarded to
quantum_kernel();"rcpp"by default.
Details
$$K_{PI}(x_i, x_j) = \alpha\, K_{quantum}(x_i, x_j) + (1 - \alpha)\, \exp\!\Bigl(-\frac{(e_i - e_j)^2}{2\,\sigma^2}\Bigr)$$
where \(e_i = y_i - \hat y_{ODE}(z_i, x_i)\) is the residual between the observed \(y_i\) and the ODE-predicted value at depth \(z_i\). The output is PSD for any \(\alpha \in [0, 1]\).