are the potential energy and pressure (long-time averages) of the system. We take the Lennard-Jones 12-6 potential as the low-fidelity input function: $$\label{eq:LJ}
\phi_0 = 4\epsilon((\sigma/r)^{12}-(\sigma/r)^6)$$ and test the ability of FUQ to correct the prediction of the QoI for a family of high-fidelity potentials. The parameters for the low-fidelity LJ potential are designed to roughly describe copper $(\sigma=2.315$Å, $\epsilon=0.167 eV)$ [@Wolf1992], fit to the bulk melt temperature and room temperature lattice constant. We note that this is not an accurate potential for Cu as it ignores important many body effects critical to describe elastic constants and defect energetics. Also notable is the discrepancy of the liquid densities as compared to experiment [@Cahill1962]. However, the goal of the paper is to demonstrate the applicability of FUQ to an MD problem. In this spirit, the high-fidelity potentials are similarly designed for demonstration purposes only and do not represent a more accurate representation of an actual material. We construct seven pair potentials by additively modifying the LJ potential with sine functions in Table 1. This family of functions will be denoted Sine 1 to Sine 7.

  Name     Modification function
  -------- ------------------------------------------
  Sine 1   $0.44+0.46\sin(0.17(24.2+r))$
  Sine 2   $-0.47\sin(-0.15(14+r))\cdot \exp(-r)$
  Sine 3   $0.07\sin(1.2(-1.2+r))/r^2$
  Sine 