Deep Learning Evidence for Global Optimality of Gerver's Sofa

TL;DR

The moving sofa problem asks for the maximal area of a planar shape that can navigate an $L$‑shaped corridor of unit width. The authors deploy two neural‑network–driven strategies to test Gerver's conjecture that his $A^*\approx 2.2195$ sofa is globally optimal: (i) a physics‑informed continuous‑area optimization using a differentiable waterfall algorithm to compute area as a function of a nonmonotone corridor movement parameterized by $x_p(t), y_p(t), \alpha(t)$, and (ii) a neural implementation of the Kallus–Romik upper bound via discrete angles and rotation centers. Their extensive experiments show that $>90\%$ of the PINN trials converge near Gerver's area, with a mean of $A\approx 2.219482$ and a refinement to $2.219515$ by L‑BFGS, while the angle‑based upper bound tightens to $2.3337$ for five angles and converges to Gerver's value from above as the number of angles grows (achieving $0.003\%$ relative error at $n=10000$). Together, these results provide strong computational evidence for the global optimality of Gerver's sofa and demonstrate a powerful combination of physics‑informed optimization and discrete angle upper bounds for tackling complex geometric optimization problems.

Abstract

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.

Figures (9)

• Figure 1: The moving sofa problem proposed by Leo Moser moser1966problem in 1966 and some lower bounds. The best-known two are found respectively by John Hammersley hammersley1968enfeeblement and Joseph Gerver gerver1992moving. The point markers separate the sections formed by different contact mechanisms (i.e., with either the inner corner, the walls, or the wall envelopes of the corridor).
• Figure 2: Geometry of the moving sofa problem. The movement of the corridor is described by the trajectory of its inner corner $P$, as denoted by $p$, and the rotation angle $\alpha$. The four walls form the four families of lines: $l_\text{ih}$, $l_\text{iv}$, $l_\text{oh}$ and $l_\text{ov}$, with the subscripts showing their initial positions (i for inner, o for outer, h for horizontal and v for vertical). Their envelopes are respectively $e_\text{ih}$, $e_\text{iv}$, $e_\text{oh}$ and $e_\text{ov}$. In this example, $p$ is a semi-ellipse with its major and minor lengths being respectively 1.8 and 1.1, leading to some complexities in the envelopes, as highlighted by the magnified windows.
• Figure 3: The waterfall algorithm for area calculation. Left: The waterfall is tested on two hand-drawn curves from both below and above. Right: The waterfall is applied to the curves formed by the corridor movement in Figure \ref{['fig:geo']}. Water sources are placed along the horizontal line in the middle, with the arrows indicating the falling direction.
• Figure 4: Our physics-informed network architecture and loss function. $\mathscr{F}$'s are unconstrained ReLU-based FCNs, and AD stands for automatic differentiation.
• Figure 5: Function initialization with network weights sampled from scaled uniform distributions. We sample weights and biases from $\mathcal{U}\left(-s\sqrt{k}, s\sqrt{k}\right)$, where $k$ is the reciprocal of input size and $s$ a number we obtain for each function by trial and error until its admissible range is safely covered from above, as indicated by the dashed lines. Note that $s=1$ is PyTorch default.