  Currently, our general approach to retrieving molecular structures from
ultrafast gas-phase diffraction heavily relies on complex ab initio electronic
or vibrational excited state simulations to make conclusive interpretations.
Without such simulations, inverting this measurement for the structural
probability distribution is typically intractable. This creates a so-called
inverse problem. In this work, we develop a broadly applicable method that
addresses this inverse problem by approximating the molecular frame structure
$|\Psi(\boldsymbol{R}, t)|^2$ distribution independent of these complex
simulations. We retrieve the vibronic ground state $|\Psi(\boldsymbol{R})|^2$
for both simulated stretched NO$_2$ and measured N$_2$O. From measured N$_2$O,
we observe 40 mAngstroms coordinate-space resolution from 3.75 inverse
Angstroms reciprocal space range and poor signal-to-noise, a 50X improvement
over traditional Fourier transform methods. In simulated NO$_2$, typical to
high signal-to-noise levels predict 100--1000X resolution improvements, down to
0.1 mAngstroms. By directly measuring the width of $|\Psi(\boldsymbol{R})|^2$,
we open ultrafast gas-phase diffraction capabilities to measurements beyond
current analysis approaches. This method has the potential to effectively turn
gas-phase ultrafast diffraction into a discovery-oriented technique to probe
systems that are prohibitively difficult to simulate.
