Model Discrepancy Function
The discrepancy function represents the systematic gap between a calibrated model and reality, distinct from random noise and parameter error.
Definition
Discrepancy delta(x) = E[reality(x) - model(x, theta*)] is the expected structural error of a model even at its best-fit parameters theta*. It is a deterministic function of the inputs, unlike observation noise which is random and mean-zero. Modeling delta explicitly is what lets calibration avoid contaminating parameter estimates with model error.
Representations
- Gaussian process with a smoothness prior, the most common choice
- Basis expansion (polynomials, splines) when the error's form is known
- Zero, only when the model is trusted to be structurally correct
Estimation
In the Kennedy-O'Hagan framework delta is inferred jointly with the calibration parameters from residuals between observations and model predictions. Because a flexible delta can absorb almost any misfit, priors constraining its magnitude and length scale are essential; otherwise the model term becomes irrelevant and calibration parameters drift to arbitrary values.
Interpretation as a diagnostic
A discrepancy that is small and unstructured suggests the model is adequate. A discrepancy with clear spatial or parametric structure points to missing physics, for example an unmodeled loss channel or an incorrect boundary condition. Reading delta as a map of where the model fails is often more valuable than the calibration itself.
Extrapolation risk
Discrepancy learned in one region of input space does not reliably transfer to another. Predictions outside the calibration domain inherit large discrepancy uncertainty, which is the correct behavior: the model has not been tested there. For fusion design this discipline keeps a reduced model honest when pushed toward operating points, such as the Hyperion breeder's frozen high-current regime, that lie beyond the data used to calibrate it.