  The inverse problem of statistical mechanics is an unsolved, century-old
challenge to learn classical pair potentials directly from experimental
scattering data. This problem was extensively investigated in the 20th century
but was eventually eclipsed by standard methods of benchmarking pair potentials
to macroscopic thermodynamic data. However, it is becoming increasingly clear
that existing force field models fail to reliably reproduce fluid structures
even in simple liquids, which can result in reduced transferability and
substantial misrepresentations of thermophysical behavior and self-assembly. In
this study, we revisited the structure inverse problem for a classical Mie
fluid to determine to what extent experimental uncertainty in neutron
scattering data influences the ability to recover classical pair potentials.
Bayesian uncertainty quantification was used to show that structure factors
with random noise smaller than 0.005 to $\sim30$ A$^{-1}$ are required to
accurately recover pair potentials from neutron scattering. Notably, modern
neutron instruments can achieve this precision to extract classical force
models to within approximately $\pm$ 1.3 for the repulsive exponent, $\pm$
0.068 A$^{-1}$ for atomic size, and 0.024 kcal/mol in the potential well-depth
with 95\% confidence. Our results suggest the exciting possibility of improving
molecular simulation accuracy through the incorporation of neutron scattering
data, advancement in structural modeling, and extraction of model-independent
measurements of local atomic forces in real fluids.
