Nonlinear Multi-Fidelity GP
When the relationship between fidelity levels is nonlinear, a deep-structured Gaussian process learns the mapping instead of assuming a constant scaling.
Beyond linear scaling
Autoregressive co-kriging assumes f_hi = rho * f_lo + delta, a linear relationship with a single scaling coefficient. Many physical systems violate this: a low-fidelity model may capture a trend that saturates, inverts, or shifts phase relative to the truth. Perdikaris and colleagues introduced nonlinear autoregressive Gaussian processes (NARGP) that replace the linear map with a learned nonlinear function.
The construction
NARGP models f_t(x) = g_t(x, f_{t-1}(x)), where g_t is itself a Gaussian process whose inputs include both the design variables and the posterior of the lower-fidelity output. A structured kernel factorizes into a part acting on x and a part acting on the propagated lower-fidelity prediction, letting the model discover space-dependent, nonlinear cross-fidelity relationships.
Uncertainty propagation
Because the lower-fidelity output enters as an uncertain input, its predictive distribution must be propagated through g_t. This is done by Monte Carlo sampling or moment matching, which makes prediction more expensive than linear co-kriging but preserves calibrated variance. The propagated uncertainty naturally inflates in regions where the nonlinear map is poorly constrained.
When to prefer it
- The scatter plot of high-fidelity vs low-fidelity output is curved, not a straight line
- A fitted linear rho is unstable across subsets of the data
- Different regions of input space show different cross-fidelity behavior
Cautions
Nonlinear multi-fidelity models have more free structure and can overfit with very few high-fidelity points. Validate with held-out high-fidelity runs and inspect whether the added flexibility actually reduces predictive error. If a linear co-kriging model already explains the cross-fidelity relationship, prefer it for its simpler, more identifiable parameters.