potential energy and pressure predicted with the low-fidelity LJ potential assuming a more accurate function is available. As discussed in subsection 3.1, a family of seven high-fidelity potentials were created by modifying the LJ potential with sine functions; see Figure \[fig:potentials\] and Table \[tbl:sine\]. These results are discussed in sub-section \[sec:resultsSine\]; the results for the Morse potential are discussed in sub-section \[sec:resultsMorse\].

Corrections for sine modified potentials {#sec:resultsSine}
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In order to use Eq. \[eq:correct\] we need, in addition to the functional derivative, the discrepancy function $(\phi_{HF} (r) - \phi_{LF} (r))$, the difference between the high and low-fidelity potentials. Figure \[fig:correct\] illustrates the functional derivative (a), the discrepancy (b), and the product of the two, the functional error (c), each as a function of interatomic distance for one case, the ambient liquid with modified potential Sine 1. These results are shown for each physical case and potential in the supplementary material. Note that the functional error goes to zero both for small and large values of $r$; at small $r$ it goes to zero following the functional derivative (due to steep repulsion that keeps atoms from coming close to one another), while for large distances the discrepancy goes to zero 