  Fictitious identical particle thermodynamics has emerged as a powerful tool
to overcome the fermion sign problem, enabling highly accurate simulations of
one thousand fermions in warm dense matter (T. Dornheim et al., J. Phys. Chem.
Lett. 15, 1305 (2024)). However, inferring the thermodynamic properties of
Fermi systems from a large number of exact numerical simulations of the bosonic
sector still poses subtle challenges, especially in the regime of high quantum
degeneracy and in the presence of phase transitions. In this work, we
demonstrate that physics-informed neural networks (PINNs), trained on data from
extensive and sign-problem-free numerical simulations of the bosonic sector,
offer a valuable means to infer the thermodynamic properties of Fermi systems.
PINNs can play a particularly crucial role in capturing phase transitions. To
illustrate the methodology of fictitious identical particles combined with
PINNs for simulating the thermodynamics of Fermi systems, we explore its
application in realistic scenarios, including ultracold Fermi gases in periodic
potentials, and phase transitions of pair condensation formed in the unitary
limit in a three-dimensional harmonic trap. For the spatially continuous
Fermi-Hubbard model, we efficiently and reliably simulated hundreds of fermions
here. For the Fermi gas in the unitary limit, based on the fictitious identical
particle combined with PINNs, our approach confirms the universal result of the
critical temperature with the increasing of the number of fermions, and is
consistent with the experimental observations.
