Deep Learning Evidence for Global Optimality of Gerver’s Sofa

Abstract

The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver’s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks’ approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver’s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our “waterfall” algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver’s solution. The second approach applies discrete optimization to the Kallus–Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver’s area from above, indicating that no larger sofa exists.

1. Introduction

Among the unsolved problems in geometry, the moving sofa problem stands out for its apparent simplicity. Formally proposed by Leo Moser [1], it asks for the two-dimensional shape of the largest area that can be maneuvered through an L-shaped corridor with unit width (see Figure 1). Beyond trivial shapes such as the square and half disk, the best-known lower bounds are those found by John Hammersley [2] and Joseph Gerver [3], as shown in Figure 1. Hammersley’s sofa, presented in a problem set for school and college students, has an area of $\pi /2+2/\pi \approx 2.2074$. It is the largest among the family whose touch point with the inner corner of the corridor forms a semi-circle; he also showed an upper bound of $2\sqrt{2}\approx 2.8284$. With an area slightly larger than $2.2195$, Gerver’s sofa is the largest known to date, composed of 18 curve sections (as partitioned by their analytical expressions). In a reasonably implicit way, Gerver proved that his solution is a local maximum and conjectured that it is the unique global maximum.

Dan Romik [4] advanced Gerver’s analysis by introducing a set of six differential equations along with their general solutions. These solutions, under prescribed contact mechanisms through five phases of rotation, land on Gerver’s sofa in a natural and explicit way. Using the same method, Romik also discovered a lower bound of the ambidextrous sofa that can pass through a double-cornered corridor. A numerical study [5] was carried out based on Romik’s formulation. Yoav Kallus and Dan Romik [6] provided a new perspective on the problem. They proved an upper bound of the maximum area through discrete optimization of the rotation center at a finite sequence of angles. Under a seven-angle setup, they discovered a new upper bound of the area of 2.37, along with a new lower bound of the least-required rotation angle for maximum area production, approximately $81.2°$. We will delve into their findings in greater detail in Section 2.3.

The past decade has witnessed groundbreaking advancements in deep learning, transforming numerous fields of human activity (e.g., computer vision [7], natural language processing [8], healthcare [9], and environmental sciences [10]). This impact extends to mathematical research, a domain increasingly referred to as AI4Math. In the realm of constructing examples or counterexamples—such as discovering new configurations for the moving sofa problem—AI has produced results that surpass traditional human efforts. Notable achievements include the development of faster algorithms for matrix multiplication [11], the discovery of larger cap sets in high dimensions [12], and the identification of more complex knots with specific topological properties [13]. Here, we aim to explore the moving sofa problem through the lens of deep learning. One of the core methodologies we will employ is physics-informed neural networks (PINNs). Originating in the 1990’s [14,15] and flourishing within the contemporary deep learning landscape [16,17], PINNs are primarily designed for solving and inverting differential equations. The central concept involves training neural networks not only with data but also by incorporating the _target equations along with their initial and boundary conditions. This approach enhances accuracy and generalization while reducing the amount of training data required. We will adhere to the principles of PINNs to address the differential terms involved in sofa area calculation, and the initial conditions of sofa movements. By doing so, we aim to achieve a more precise and efficient solution to the moving sofa problem.

In this paper, we approach the moving sofa problem from an optimization standpoint, employing neural networks to parameterize a function space to identify optimal sofa movements or upper bounds. This method harnesses the universal approximation capability of neural networks and modern optimizers’ enhanced, albeit still evolving, proficiency to avoid local minima. Specifically, we take advantage of recent developments in invexity optimization [18,19] to ensure that the employed neural networks will map only global optima. Here, an invex function is a mapping where any critical point is a global minimizer. While preliminary attempts from the community exist (e.g., on GitHub and Stack Overflow), these efforts often fail to consider geometric constraints accurately and lack comprehensive insights into optimality. A significant challenge we have overcome is the computation of the sofa area. Our solution ensures that the area is calculated not only with accuracy and robustness but also in a manner that supports differentiation, as required for backpropagation.

The remainder of the paper is organized as follows. In Section 2, we present two problem formulations—one optimizing sofa area through corridor movement and the other approaching the Kallu-s-Romik upper bound [6]. Section 3 describes our deep learning methodology, highlighting the generality of the function space and diverse initialization to reduce inductive bias. Finally, Section 4 presents our findings, reinforcing the global optimality of Gerver’s sofa. To the best of our knowledge, this is the first study to leverage deep learning in addressing the moving sofa problem. It represents a substantial advancement by integrating geometric exactness with differentiable optimization techniques.

2. Problem Definition and Formulations

2.1. Sofa Formation

The moving sofa problem can be understood as determining the movement of the L-shaped corridor in ${\mathbb{R}}^2$ to maximize the area not swept by its four walls. We describe its movement using the trajectory of its inner corner, denoted by p, and the rotation angle $\alpha$, as illustrated in Figure 2. This representation allows us to describe the corridor’s position and orientation at any moment. An interactive visualization, assuming an elliptical trajectory for p, has been contributed by the community [20].

Previous studies using continuous optimization have considered as functions of $\alpha$ [3,4,5], thereby restricting the movement to a monotonic rotation. To overcome this limitation, we introduce a pseudo-time , which allows us to describe the movement as follows:

(1)

The above initial conditions at $t=0$ are trivial, addressed by the translation and rotation invariance of ${\mathbb{R}}^2$. A non-trivial condition is

$$\alpha \left(1\right)>arcsin{\displaystyle \frac{84}{85}}\approx 81.2°,$$

(2)

that is, any moving sofa shape of the largest area must undergo rotation by an angle of at least $81.2°$ [6]. We will later use this condition in our loss function. The parameterization in Equation (1) only requires that $x_p\left(t\right)$, $y_p\left(t\right)$ and $\alpha \left(t\right)$ are of class $C^0$, allowing for independent translation and rotation that are non-monotonic, non-symmetric, and non-differentiable.

The movement of the corridor will determine the following four families of lines, named by their initial position at $t=0$ (see Figure 2):

(3)

Based on simple calculus, their envelopes can be shown as

(4)

where the primes denote derivatives with respect to t. We assume that $x_p\left(t\right)$, $y_p\left(t\right)$, and $\alpha \left(t\right)$ are piecewise differentiable (piecewise $C^1$) so that any non-differentiable points can be excluded from the aforementioned envelopes. This assumption is justified, as it is unlikely that the largest sofa could contain some fractal sections. If it does indeed, such sections are inherently undetectable by methods based on analytical parameterization, including deep learning approaches.

2.2. Differentiable Area Calculation

The resultant area is bounded from below by p, $e_{\mathrm{ih}}$, $e_{\mathrm{iv}}$, and $y=0$, depending on which one is on top at any given point. Similarly, it is bounded from above by $e_{\mathrm{oh}}$, $e_{\mathrm{ov}}$, and $y=1$, depending on which one is at the bottom. However, it cannot be formulated as a simple integral because these curves can be self-intersecting and intersect at multiple points, as indicated by the zoomed-in windows in Figure 2. Note that Figure 2 only shows a simple case where the trajectory is elliptical; the envelopes can become intricate for arbitrary movements. For example, Figure 3 shows the envelopes created by an irregular movement. This complexity contributes to why the moving sofa problem remains unsolved.

To compute the area robustly for any movement, we developed the waterfall algorithm, inspired by the watershed algorithm for image segmentation [21,22]. The watershed algorithm is a region-based image segmentation method that treats the image as a topographic surface, where pixels represent elevation; it identifies “watershed lines” to separate distinct regions. We adapt the concept of using “water” to identify surfaces from above and below. Intuitively, a dense series of water sources aligned horizontally are placed high above, generating a waterfall that touches the intersecting curves to form the upper limit for area quadrature. The lower limit is determined similarly, but with the water flowing upward from below. Our implementation ensures differentiability of the resultant area with respect to the curve positions, which is essential for backpropagation. We also parallelize the water source through sensitization to achieve faster computation. Some examples are given in Figure 4.

We optimize the calculated sofa area as a function of corridor movement, where the _targets $x_p\left(t\right)$, $y_p\left(t\right)$, and $\alpha \left(t\right)$ are modeled using neural networks, as detailed in Section 3. The corresponding results are presented in Section 4.1.

2.3. Kallus–Romik Upper Bound

Yoav Kallus and Dan Romik [6] established an upper bound for the maximum sofa area by optimizing the rotation center at a finite sequence of rotation angles. We provide a brief review of their theory and discuss some limitations of their numerical algorithm.

Let $L_{\alpha }\left(\mathit{u}\right)$ denote the set formed by rotating the corridor by an angle $\alpha$ around the center :

(5)

Let denote the set formed by rotating the vertical strip by angles ${\beta }_1$ and ${\beta }_2$, respectively, and taking the union of these two rotations, resulting in a butterfly-shaped region:

(6)

Let $\lambda$ denote the area measure on ${\mathbb{R}}^2$, and ${\lambda }^{*}\left(\mathcal{X}\right)$ denote the maximal area of any connected component $\mathcal{X}\subset {\mathbb{R}}^2$. Then, given a finite sequence of angles, , and their corresponding rotation centers, , and with ${\beta }_1\le {\beta }_2$, Kallus and Romik [6] first defined the following area:

(7)

where is the horizontal strip. Taking the supremum of the above area with respect to the centers, they further obtained

(8)

which they ultimately proved to be an upper bound of the maximum sofa area. Note that ; namely, when ${\beta }_2=\pi /2$, the value of ${\beta }_1$ does not affect the resultant area. Therefore, the end case of $G_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k}^{{\beta }_1,\pi /2}$ can simply be denoted by $G_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k}$. The above-defined sets, $L_{\alpha }\left(\mathit{u}\right)$ and , and their induced $g_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k}^{{\beta }_1,{\beta }_2}$ are illustrated in Figure 5, based on a small number of angles.

For a sofa shape that moves around the corner while rotating continuously and monotonically from 0 to some angle $\beta \in [0,\pi /2]$, define its maximum area as $A\left(\beta \right)$. The global maximum area is then $A_{max}={sup}_{0\le \beta \le \pi /2}A\left(\beta \right)$. The most relevant results from Kallus and Romik [6] are summarized as follows:

• for any and satisfying ${\alpha }_1<{\alpha }_2<\cdots <{\alpha }_k\le {\beta }_1\le \beta \le {\beta }_2\le \pi /2$, one has [6] [Proposition 4 (iii)]

  $$A\left(\beta \right)\le G_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k}^{{\beta }_1,{\beta }_2},$$

  (9)

  and [6] [Proposition 4 (ii)]

  $$A_{max}\le G_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k};$$

  (10)
• given an integer $n\ge 3$, let ${\gamma }_j={\displaystyle \frac{j}{n}}{\displaystyle \frac{\pi }{2}}$, $j=1,2,\cdots ,n$;  [6] [Theorem 5] asserts that

  $$A_{max}=\underset{n\to \infty }{lim}\underset{k=1,2,\cdots ,\lceil n/3\rceil }{max}{G}_{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_{n-k-1}}^{{\gamma }_{n-k},{\gamma }_{n-k+1}},$$

  (11)

  which provides a way for constraining $A_{max}$ asymptotically from above.

The above results are rigorous and elegant from a theoretical perspective. However, the numerical algorithm they present for discrete optimization of the rotation centers, , seems to have two shortcomings:

• Their algorithm does not yield $G_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k}^{{\beta }_1,{\beta }_2}$ but instead provides another upper bound whose tightness is unproven; though this computed one is also an upper bound of $A\left(\beta \right)$, there can be space for improving the tightness if $G_{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k}^{{\beta }_1,{\beta }_2}$ is evaluated directly.
• Based on rational programming, their algorithm is computationally expensive and unscalable, which can only tackle a small number of angles. Consequently, Equation (11) or their Theorem 5 remains unexploited for constraining the actual value of $A_{max}$.

These shortcomings motivate our work reported in Section 4.2 and Section 4.3.

3. Deep Learning Methodology

3.1. Fully Connected Networks

Neural networks, particularly fully connected networks (FCNs), play a central role in many machine learning tasks due to their ability to approximate complex functions. An FCN consists of multiple layers of nodes, where each node applies a weighted sum to its inputs, followed by an activation function to introduce non-linearity. This structure allows FCNs to act as universal approximators, capable of representing a broad range of continuous functions [23,24]. In deep learning, such models are trained to optimize parameters that minimize a loss function, guiding the model towards accurate predictions. The rectified linear unit (ReLU) activation function, commonly used in FCNs, introduces piecewise linearity [25]. This property makes ReLU-based FCNs ideal for exploring a broad function class, as required by the optimization problems discussed here.

In Appendix A, we provide a theoretical review of FCNs, including their formulation, universal approximation ability, and characterization of invexity. Universal approximation increases the likelihood that the global maximum is included in the solution space. Invexity enhances the probability that the first-order, gradient-based optimizers used in deep learning, featuring some capability to escape from local minima, can successfully locate the global minima. Together, these properties reinforce the appeal of our conclusions from extensive function search.

3.2. Network Parameterization of Corridor Movement

Distinct from typical deep learning tasks focused on data fitting, our application is not primarily concerned with the accuracy or generalization of a single best model. Instead, our objective is to support the global optimality of Gerver’s sofa by demonstrating that multiple models from our over-parameterized function space converge to Gerver’s lower bound. Consequently, we want our function space, or hypothesis class, to be as general as possible. We parameterize $x_p\left(t\right)$, $y_p\left(t\right)$, and $\alpha \left(t\right)$ by three independent FCNs with ReLU as the activation function. FCNs are chosen for their universal approximation capabilities [23,24], and ReLU is selected for its ability to provide piecewise linear parameterization [25,26]. It is worth mentioning that using $C^{\infty }$ activation functions such as tanh and softplus instead of ReLU, would limit the search space to $C^{\infty }$ functions. Though they can lead to faster convergence to Gerver’s sofa, our approach favors piecewise $C^1$ functions for broader generality. As explained, piecewise $C^1$ is likely the most general class of functions that can be explored using non-fractal methods.

Specifically, we mathematically show that our proposed neural networks map to global optima (see Appendix A). We remark that this theoretical result plays a crucial role in addressing the moving sofa problem. It ensures that any critical point corresponds to a global optimum (invexity definition [27]), a necessary condition for achieving the global optimality of Gerver’s sofa. In Section 4, we will provide numerical validation for this optimality by illustrating the landscapes of our network parameterization.

In general, neural network optimization, while gradient-based, does not guarantee convergence to a global minimum. However, contemporary optimizers like Adam [28,29], which incorporate adaptive learning rates and momentum, enhance the probability of escaping local minima. Additionally, the likelihood of attaining the global minimum increases if multiple approximators, initialized diversely, converge to the same local minimum. In this context, “diverse initialization” extends beyond mere variation in weights and biases; it implies that approximators approach their resultant minima from distinct trajectories in the parameter space. This requires meticulous scaling of the weight sampling distribution. For instance, our _target function $\alpha \left(t\right)$ lies within , whose approximators should hence be initialized to safely cover . This consideration diverges from typical machine learning practices. Figure 6 illustrates a subset of our initialization samples.

3.3. Physics-Informed Architecture

Our neural networks are physics informed, leveraging principles outlined in the recent literature [16,17]. Firstly, as indicated in Equation (4), the computation of the envelopes—and consequently the area—depends on the derivatives of the learned functions ($x_p^{\prime }$, $y_p^{\prime }$, and ${\alpha }^{\prime }$), which are efficiently calculated via automatic differentiation (AD). Secondly, we adhere to the initial conditions specified in Equation (1) directly within the network architecture, while the non-trivial condition in Equation (2) is incorporated into the loss function due to its unequal nature. Additionally, it is straightforward to see that a movement involving $x_p>0$, $y_p<0$, or $\alpha <0$ is forbidden by geometry. Combing all of these considerations, the functions describing a corridor movement are approximated by

(12)

where ${\mathcal{F}}_{x_p}$, ${\mathcal{F}}_{y_p}$, and ${\mathcal{F}}_{\alpha }$ are unconstrained FCNs. With the area A computed by our waterfall algorithm based on $x_p\left(t\right)$, $y_p\left(t\right)$, and $\alpha \left(t\right)$, the following loss function is adopted for backpropagation:

(13)

where the ReLU deactivates the penalizing term once $\alpha \left(1\right)$ exceeds $81.2°$. The complete architecture is summarized in Figure 7.

For the Kallus–Romik upper bound, we use unconstrained FCNs to parameterize the functions $u_1\left(\alpha \right)$ and $u_2\left(\alpha \right)$; that is, $u_{1,2}\left(\alpha \right)={\mathcal{F}}_{u_{1,2}}\left(\alpha \right)$, with no initial conditions and sign enforcement. The waterfall algorithm is applied directly on the four rays in $L_{\alpha }\left(\mathit{u}\right)$ and the two lines in , instead of on envelopes. Note that envelopes do not exist because the upper bound is defined with discrete angles. These two differences indicate that our neural networks are no longer informed by any “physics”, which is consistent with the notion that the Kallus–Romik upper bound is not necessarily associated with a valid sofa movement [6].

4. Results

4.1. Physics-Informed Area Optimization

We discretize the interval into 2000 points, with 10,000 sources allocated to the waterfall. This configuration is determined through iterative refinement, ensuring that the area converges to the fourth decimal place when tested on Gerver’s shape. We conduct a comprehensive exploration of 6000 runs, incorporating 1000 random seeds for weight initialization, 3 initial learning rates (${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$, halved every 2000 epochs), and 2 network architectures (2 hidden layers of size 256 and 3 hidden layers of size 128). These architectures are selected to practically apply Theorem A2 (Appendix A) under the full row-rank hypothesis. Moreover, since we use ReLU activation, which is strictly increasing in its active domain, the assumptions of Theorem A2 are inherently satisfied. Each run consists of 10,000 epochs, using the Adam optimizer [28]. To ensure high-precision area calculations, we utilize float64, resulting in a runtime of approximately two hours on the CPU, with minimal performance gain from GPU acceleration. The use of float64 is critical for reliable statistical results, as float32 occasionally produces areas that slightly exceed 2.2195.

In our analysis of the 6000 experimental runs, approximately 8.5% resulted in a vanishing area, primarily occurring when the learning rate is highest. After excluding these outliers, the remaining trials exhibit strong convergence, consistently approaching a small neighborhood around Gerver’s solution, with an area close to 2.219532 [30]. The observed areas follow a normal distribution, $\mathcal{N}(\mu ,{\sigma }^2)$, where $\mu =2.219482$ and $\sigma =0.000011$. The area values range from 2.219459 to 2.219502. Further refinement with 1000 epochs of fine tuning using the L-BFGS algorithm [31] increases the maximum area to 2.219515. Due to resource constraints, L-BFGS is not applied to other models.

We further extend our experiments to include deeper and wider architectures, $C^{\infty }$ activation functions, and diverse weight sampling distributions for rigorous verification. The models using the tanh activation function yield a strikingly similar distribution while exhibiting accelerated convergence in the initial phases of training, before the area approaches the vicinity of 2.219.

Finally, to visually represent our findings, Figure 8 illustrates the Hessian-based landscape of the sofa area for one of our representative models. This visualization offers compelling evidence for the global optimality of Gerver’s sofa within our network-parameterized function space.

4.2. Kallus–Romik Upper Bound with Many Angles

The computational cost of our deep learning workflow (waterfall and backpropagation) scales with the number of rotation angles in the Kallus–Romik upper bound, allowing us to directly explore the convergence theorem in Equation (11). We increase the number of angles (n) from 10 to 10,000, incremented by 10 between , by 100 between and by 500 afterward, making a total of 53 values for n. For each n, Equation (11) requires $\lceil n/3\rceil$ training instances, which are still computationally expensive for large n’s. While Kallus and Romik [6] have shown that $81.2°$ is the minimum angle, the corridor must rotate for maximum area production; all current evidence suggests that a rotation of $\pi /2$ should take place [3,4,5,6], including our results reported in Section 4.1. Therefore, we will vary model initialization only for $k=1$ (rotation by $\pi /2$) with 20 random seeds (i.e., using 1 seed for $k>1$). The other hyper-parameters (learning rates, architectures, and epochs) remain the same as those in Section 4.1. The most expensive run for n = 10,000, with 10,000 sources for the waterfall and 10,000 epochs, takes approximately 10 CPU hours to complete.

Our training results show that, for all values of n’s, the maximum $G_{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_{n-k-1}}^{{\gamma }_{n-k},{\gamma }_{n-k+1}}$ is achieved when $k=1$, supporting of the conjecture that the largest sofa necessitates a full rotation by $\pi /2$. We can then focus our attention on $G_{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_{n-2}}^{{\gamma }_{n-1},{\gamma }_n}$ (i.e., $k=1$). As shown in Figure 9, $G_{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_{n-2}}^{{\gamma }_{n-1},{\gamma }_n}$ found by our models converges asymptotically to Gerver’s area from above. The relative error becomes smaller than $1\%$, $0.1\%$, and $0.01\%$ respectively at $n=30$, 300, and 2100, and finally reaches $0.003\%$ at n = 10,000. This result provides another piece of evidence for the global optimality of Gerver’s sofa, along with the findings presented in Section 4.1.

4.3. Kallus–Romik Upper Bound with Five Angles

An outstanding advantage of the Kallus–Romik upper bound is that it allows for exploring the problem at a low dimension by considering only a few angles. Kallus and Romik [6] defined the following five angles (note that they originally defined seven, but the other two are irrelevant to the sofa area):

$$\begin{array}{cc}\hfill & {\alpha }_1=arcsin{\displaystyle \frac{7}{25}}\approx 16.26°,\hfill \\ \hfill & {\alpha }_2=arcsin{\displaystyle \frac{33}{65}}\approx 30.51°,\hfill \\ \hfill & {\alpha }_3=arcsin{\displaystyle \frac{119}{169}}\approx 44.76°,\hfill \\ \hfill & {\alpha }_4=arcsin{\displaystyle \frac{56}{65}}={\displaystyle \frac{\pi }{2}}-{\alpha }_2\approx 59.59°,\hfill \\ \hfill & {\alpha }_5=arcsin{\displaystyle \frac{24}{25}}={\displaystyle \frac{\pi }{2}}-{\alpha }_1\approx 73.74°.\hfill \end{array}$$

(14)

These angles do not have a specific geometric meaning; they are derived from arcsin functions because the numerical algorithm of Kallus and Romik [6] is rational.

As we have explored the Kallus–Romik upper bound with up to 10,000 rotation angles in Section 4.2, the five-angle case holds less mathematical significance. However, its smaller scale enables us to conduct a more exhaustive model search and visualize the results in terms of the landscape with respect to the rotation centers. Both further reinforce the conclusions drawn in Section 4.2 based on many angles.

The inequalities below combine the optimization results of their algorithm with those of our neural networks for $G_{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5}$ and $G_{{\alpha }_1,{\alpha }_2,{\alpha }_3}^{{\alpha }_4,{\alpha }_5}$, with ours in the middle:

$$\begin{array}{cc}{G}_{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5}\hfill & \le 2.3337\pm 0.0000<2.37,\hfill \\ \hfill G_{{\alpha }_1,{\alpha }_2,{\alpha }_3}^{{\alpha }_4,{\alpha }_5}& \le 1.9259\pm 0.0000<2.21.\hfill \end{array}$$

(15)

Our results come from 6000 runs, using the same hyper-parameters (learning rates, architectures, seeds and epochs) as described in Section 4.1. The training converges so well that the variances almost vanish (i.e., $\pm 0.0000$ in the inequalities in Equation (15)).

Based on Equation (9), it can be shown that $A_{max}\le max(G_{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5},G_{{\alpha }_1,{\alpha }_2,{\alpha }_3}^{{\alpha }_4,{\alpha }_5})$. Therefore, our results suggest a tighter upper bound of 2.3337, corresponding to the set shown in Figure 10. The landscape of $g_{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5}$ is also shown in Figure 10, verifying the global optimality of $G_{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5}$ in our network-parameterized function space.

5. Discussion

We have provided two lines of evidence supporting Gerver’s conjecture that his 18-section sofa, with an area of approximately 2.2195, is the unique shape of maximum area that can navigate through a unit-width L-shaped corridor. First, we optimized the sofa area using PINNs, focusing on minimizing inductive bias by ensuring generality in the function space and diversity in function initialization. After training more than 6000 models, over 90% converged on Gerver’s solution. Second, we refined the Kallus–Romik upper bound [6], achieving a tighter upper bound of around 2.3337 compared to their previous estimate of 2.37. By increasing the number of rotation angles to 10,000, our training results showed that the upper bound asymptotically approaches Gerver’s area, further supporting the conclusion that no larger sofa exists.

If a larger sofa does exist, how has it remained undiscovered for nearly 60 years? We propose two plausible explanations in light of all previous attempts, including ours. First, the shape may be fractal or include a fractal section. If so, its discovery would be highly challenging, as we still struggle to estimate areas of known fractals. Second, the global maximum might have such a small catchment basin that it is almost undetectable by optimization-based methods, whether gradient-based or non-gradient-based. In either case, any parameterized function space such as neural networks, would be insufficient. Instead, geometric insights revealing a new movement pattern with contact mechanisms distinct from Gerver’s (such as those described in [4]) would be more critical than further algorithmic advancements.