  The Moving Sofa Problem, formally proposed by Leo Moser in 1966, seeks to
determine the largest area of a two-dimensional shape that can navigate through
an $L$-shaped corridor with unit width. The current best lower bound is about
2.2195, achieved by Joseph Gerver in 1992, though its global optimality remains
unproven. In this paper, we investigate this problem by leveraging the
universal approximation strength and computational efficiency of neural
networks. We report two approaches, both supporting Gerver's conjecture that
his shape is the unique global maximum. Our first approach is continuous
function learning. We drop Gerver's assumptions that i) the rotation of the
corridor is monotonic and symmetric and, ii) the trajectory of its corner as a
function of rotation is continuously differentiable. We parameterize rotation
and trajectory by independent piecewise linear neural networks (with input
being some pseudo time), allowing for rich movements such as backward rotation
and pure translation. We then compute the sofa area as a differentiable
function of rotation and trajectory using our "waterfall" algorithm. Our final
loss function includes differential terms and initial conditions, leveraging
the principles of physics-informed machine learning. Under such settings,
extensive training starting from diverse function initialization and
hyperparameters is conducted, unexceptionally showing rapid convergence to
Gerver's solution. Our second approach is via discrete optimization of the
Kallus-Romik upper bound, which converges to the maximum sofa area from above
as the number of rotation angles increases. We uplift this number to 10000 to
reveal its asymptotic behavior. It turns out that the upper bound yielded by
our models does converge to Gerver's area (within an error of 0.01% when the
number of angles reaches 2100). We also improve their five-angle upper bound
from 2.37 to 2.3337.
