of $9.02 g/cm^3$), one liquid slightly above the melt temperature at ambient pressure ($1300 K$ with density of $6.48 g/cm^3$) and another liquid at extreme temperature and pressure ($5000 K$ with density of $8.93 g/cm^3$, corresponding to $55 GPa$).

All MD simulations were performed using the LAMMPS package [@Plimpton1995]. For each high-fidelity potential we performed 16 independent simulations, each 1 ns long, sufficient for good statistical sampling of the quantities of interest. For each low-fidelity potential, from which the functional derivatives were calculated, the total sample time was 64 ns, similarly from 64 independent simulations of 1 ns. The convergence of this sampling is shown in the supplementary material.

Numerical functional derivatives and a perturbative approach for their calculation
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For the specific case of interest the functional derivative expression becomes: $$\label{eq:FD2}
\frac{\delta Q[\phi]}{\delta \phi}(r_i) = \lim_{\epsilon \to 0} \frac{Q[\phi_0(r) + \epsilon\cdot\phi'(r-r_i)] - Q[\phi_0(r)]}{\epsilon}$$ where $Q[\phi]$ denotes the average QoI – the potential energy or pressure of the system – using interatomic potential $\phi$. The unmodified LJ potential is $\phi_0$ and $\phi'$ a normalized Gaussian perturbation centered at interatomic distance $r=r_i$ with width $\sigma$.

The functional derivative in Eq. \[eq:FD2\] can be calculated by performing a set of simulations with 