in Figure \[fig:hybrid\].

$$\label{eq:hybrid}
\phi_{Hybrid} =
\begin{cases} 
      \phi_{LJ}(r) & r\geq 2.8 \\
      \phi_{LJ}(r)\cdot [B(r)+0.5] + \phi_{Morse}(r)\cdot [-B(r)+0.5] & 2.8 > r > 2.4 \\
      \phi_{Morse}(r) &  2.4 \geq r \\ 
\end{cases}$$

$$\label{eq:erf}
     B(r) = 0.5\operatorname{erf}(8(r-2.6))$$

With the hybrid potential in Eq. \[eq:hybrid\] the high-fidelity model is used only where necessary. This is similar in spirit to adaptive sampling methods in multi-scale simulations, with examples in the literature [@Knap2008; @Barton2011; @Roehm2015] and codes available: the Co-design Embedded Visco-Plasticity Proxy Application (CoEVP) [@Dorr2014] and the Co-design Heterogeneous Multiscale Method Proxy Application (CoHMM) [@RouetLeduc2014].

It is then possible to use FUQ to correct a low-fidelity simulation run with this hybrid potential to the result with the high-fidelity Morse by using the discrepancy between the two potentials and the functional derivative of the hybrid potential. The hybrid potential reduces the discrepancy and the functional approach provides an accurate correction for all cases, as shown with the white bars in Figure \[fig:results\]. The solid case correction shows an additional example of increased error, again due to poor overlap in explored phase space between the two potentials (see supplementary material). Even with these significantly different systems FUQ provides the majority of the necessary correction. Further, all 