  Describing the ground states of continuous, real-space quantum many-body
systems, like atoms and molecules, is a significant computational challenge
with applications throughout the physical sciences. Recent progress was made by
variational methods based on machine learning (ML) ansatzes. However, since
these approaches are based on energy minimization, ansatzes must be twice
differentiable. This (a) precludes the use of many powerful classes of ML
models; and (b) makes the enforcement of bosonic, fermionic, and other
symmetries costly. Furthermore, (c) the optimization procedure is often
unstable unless it is done by imaginary time propagation, which is often
impractically expensive in modern ML models with many parameters. The
stochastic representation of wavefunctions (SRW), introduced in Nat Commun 14,
3601 (2023), is a recent approach to overcoming (c). SRW enables imaginary time
propagation at scale, and makes some headway towards the solution of problem
(b), but remains limited by problem (a). Here, we argue that combining SRW with
path integral techniques leads to a new formulation that overcomes all three
problems simultaneously. As a demonstration, we apply the approach to
generalized ``Hooke's atoms'': interacting particles in harmonic wells. We
benchmark our results against state-of-the-art data where possible, and use it
to investigate the crossover between the Fermi liquid and the Wigner molecule
within closed-shell systems. Our results shed new light on the competition
between interaction-driven symmetry breaking and kinetic-energy-driven
delocalization.
