ensemble – depends on the input function, here the Lennard-Jones (LJ) two-body pair potential. We further show that the FD with respect to the LJ potential can be used to compute accurate correction to the potential energy and pressure for a family of pair potentials without re-running the simulation. This is true as long as the discrepancy between the potentials remains within reasonable bounds and the phase space explored by the system with the new potential is not significantly different from that of the original.

The remainder of the paper is as follows. Section 2 describes the functional approach to UQ followed by simulation details and methods to calculate the functional derivative numerically in Section 3. In Section 4 we describe specific examples of error correction using FUQ with low and high-fidelity models. Sections 5 and 6 discuss results and conclude the paper.

Functional uncertainty quantification
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In general, a simulation predicts a quantity of interest, $Q$, given some set of input parameters $P_i$ and constitutive functions $f_j$, themselves functions of an independent variable $z$ and input parameters $N_k$: $$\label{eq:QoI}
Q=Q({P_i},{f_j(z,{N_k})})$$ In the cases of interest here, $Q$ will be the time averaged potential energy or pressure and $f(z)$ the 