LJ potential as a function of interatomic distance used in the MD simulation.

The most common form of UQ is concerned with uncertainty in simulation outputs arising from uncertainty in the input parameters. This approach, while very valuable, ignores the fact that the functional forms used as constitutive laws are, almost invariably, approximate and lead to errors in the simulation. To quantify uncertainties with respect to input functions we utilize functional derivatives (Fréchet derivatives) of the QoI with respect to the input functions. The FD can be written in differential form as: $$\label{eq:FD1}
\frac{\delta Q[f]}{\delta f(z)}(z_i) = \lim_{\epsilon \to 0} \frac{Q[f(z) + \epsilon\cdot\delta(z-z_i)] - Q[f(z)]}{\epsilon}$$ This characterizes the functional sensitivity of the QoI with respect to the input function. The definition in Eq. \[eq:FD1\] uses the Dirac delta function as the functional variation; as we describe below, to calculate the functional derivative numerically for the MD simulations we use narrow Gaussian distributions centered at $z_i$.

Besides quantifying uncertainties in the prediction given uncertain input functions, the functional derivative can be used to correct the error that arises from the use of a low-fidelity model $(f)$ if a higher-fidelity one $(g)$ becomes available. The correction is then obtained using the product 