The Basins of Attraction of Soft Sphere Packings Are Not Fractal

Abstract

The energy landscape picture is a central tool to study many-body systems. In particular, the energy landscapes of glass-forming liquids, jammed packings, constraint satisfaction problems, or neural networks contain a plethora of minima corresponding to competing states. Due to their complexity, these landscapes resist analytical treatment and must be studied numerically. We focus on jammed soft spheres, a paradigmatic model of glasses and granulars, to expose the limitations of standard numerical methods in resolving the true structure of energy landscapes. We show that CVODE is the ODE solver with the best time-for-error trade-off, outperforming commonly used steepest-descent solvers by several orders of magnitude. Using this numerical approach, we provide unequivocal evidence that optimizers widely used in computational studies destroy all semblance of the true landscape geometry, even in moderately low dimensions. Employing a range of geometric indicators, both low- and high-dimensional, we show that earlier claims on the fractality of basins of attraction of minima originated from the use of inadequate mapping strategies. In reality, the basins of attraction of soft sphere packings are smooth structures with well-defined length scales, a result that likely extends to a much broader family of problems.

Figures (8)

• Figure 1: Slicing the energy landscape.$1350 \times 2400$ pixels in a random $2d$ plane in the configuration space of $N = 16$ disks. At each pixel, we use CVODE (top) and FIRE (bottom), to identify which basin of attraction it belongs to. Each basin is uniquely encoded by one color across both panels.
• Figure 2: Optimizers: fast but inaccurate.$(a)$ Average accuracy of algorithms, computed over $10^4$ random points. Error bars are Clopper-Pearson $95\%$ confidence intervals Clopper1934supp. Dashed lines are exponential fits, long-dashed line stretched exponential fits. $(b)$ Corresponding average computation times. Error bars are Student-T $95\%$ confidence intervals supp. $(c)-(e)$$800\times 800$-pixel slices of configuration space for $N=128$ particles for $(c)$ CVODE, $(d)$ GD, $(e)$ FIRE, and $(f)$ L-BFGS.
• Figure 3: Linear intersects of basins.$(a)$ Intersection lengths distributions obtained with L-BFGS (blue triangles) and CVODE (green squares) over $10$ lines of $10^6$ pixels for $N = 16$, in log scales. A dashed line indicates $1/\ell$ behavior. $(b)$ CDF of the distribution of $\log \ell$ obtained with CVODE by zooming $100\times$ on each basin boundary found from panel $(a)$. The dashed black line is a truncated Gaussian fit. Inset: Corresponding histogram of the pdf, the dashed line shows a kernel regression.
• Figure 4: Survival$(a)$ Survival from a random point in the landscape of $N=1024$ disks, in log-log scale. Solid black lines are stretched-exponential fits for CVODE and FIRE. Dashed gray line: saturating power-law fit to L-BFGS. Error bars are Clopper-Pearson estimates Clopper1934. $(b)$ Same curve at $\phi = 0.85$. Gray lines are power-law fits for FIRE and L-BFGS. The black line is a stretched-exponential fit to CVODE. $(c)$ Half-survival radius $R_{1/2}$ inside a single basin against distance $r$ to its minimum for $N=128$. Lines indicate proposed fits, exponential (gray) for FIRE and L-BFGS and stretched-exponential (black) for CVODE. $(d)$ Average distance $\overline{d_{min}}$ between neighboring minima using kicks of size $R$, for $N=1024$. Inset: log-scale relative error in $\overline{d_{min}}$ for optimizers vs. CVODE.
• Figure 5: Basin volumes.$(a)$ Intensive free energies $F_0/N$ across methods, against $N$, each averaged over the same 5 basins for each $N$. $(b)-(c)$ Example densities of states (DOS) for one basin per $N$, for $(b)$ CVODE, and $(c)$ FIRE. $(d)$ Averaged accuracy over $\mathcal{O}(10^5)$ FIRE samples used, against their distance to the minimum, in semi-log scale. Dashed lines are exponential fits $y = C \exp(-\lambda x)$ of each curve. Inset: best decay rate $\lambda$ against $N$ in log-log, with a dashed power-law $\lambda \sim N^{0.6}$. $(e)$$\log_{10}$ of the ratio between the CVODE DOS of the basins in $(b)-(c)$ and that of a hyperball in $N(d-1)$ dimensions. Dashed lines are exponential fits. Inset: decay rate $\mu$ against $N$ in log-log, with a dashed power-law $\mu \sim N^{0.9}$.