Phase-Space Engineering and Collective Dynamics in Memcomputing

TL;DR

This work investigates how hyper-parameters govern phase-space geometry and collective dynamics in digital memcomputing machines (DMMs) solving combinatorial problems. Using numerical simulations of a prototypical DMM for planted-solution 3-SAT, it maps viable regions in the memory-fast timescale space defined by $τ_x > τ_v$ (with $τ_x oughly 1/β$) and analyzes avalanche statistics, instantons, and time-to-solution distributions. The authors show a wide region where solution search is efficient due to memory-induced collective dynamics, while too-fast or too-slow memory deteriorates performance via breakdown of collectivity or noise-driven anti-instantons; the time-to-solution (TTS) distributions follow an inverse Gaussian form across regimes. The findings provide practical guidance for tuning memory-related hyper-parameters to maximize phase-space navigation and computational efficiency in memcomputing hardware, highlighting the robustness of the underlying phase-space topology to parameter variations.

Abstract

Digital Memcomputing machines (DMMs) are dynamical systems with memory (time non-locality) that have been designed to solve combinatorial optimization problems. Their corresponding ordinary differential equations depend on a few hyper-parameters that define both the system's relevant time scales and its phase-space geometry. Using numerical simulations on a prototypical DMM, we analyze the role of these physical parameters in engineering the phase space to either help or hinder the solution search by DMMs. We find that the DMM explores its phase space efficiently for a wide range of parameters, aided by the system-wide correlations in their fast degrees of freedom that emerge dynamically due to coupling with the (slow) memory degrees of freedom. In this regime, the time it takes for the system to find a solution scales well as the number of variables increases. When these hyper-parameters are chosen poorly, the system navigates its phase space far less efficiently. However, we find that, in many cases, collective behavior persists even when the phase-space exploration process is inefficient. This behavior only disappears if the memories are made to evolve as quickly as the fast degrees of freedom. This study points to the important role of memory and hyper-parameters in engineering the DMMs' phase space for optimal computational efficiency.

Figures (4)

• Figure 1: (a) State space diagram in hyper-parameter space $\{\beta, \zeta\}$, showing the maximum size 3-SAT ($\alpha_r = 4.3$) instance $N_{max}$ that a DMM can solve within a fixed number of integration steps $T_{median}$ with fixed $dt_0 = 1.0$. The number of logical variables, $N$, is increased by $10$ at a time until $T_{median}$ exceeds $10^6$ steps. We find a wide region in phase space in which moderately large $N$ instances ($\sim400-600$) can be solved quickly without tuning $dt_0$ (which scales the adaptive time step). Colored dots and stars correspond to parameter choices depicted in (b). (b) Scalability plots of the median number of steps to find a solution $T_{median}$ as a function of number of logical variables $N$. For two sets of parameters ($\{ \beta, \zeta \} = \{ \beta_{opt}, \zeta_{opt} \}= \{ 79, 0.063\}$ and $\{ 6.0 \beta_{opt}, 1.5 \zeta_{opt} \}$), the $T_{median}(N)$'s are fitted well by polynomials ($\sim N^{1.65}$ and $\sim N^{3.18}$, respectively). Outside the phase of "good" parameters, $T_{median}(N)$'s are fitted well by exponentials ($\sim e^{0.12N}$, $\sim e^{0.07N}$, and $\sim e^{0.04N}$ for $\{ \beta, \zeta \} = \{ \beta_{opt}, \zeta_{opt}/10 \}$, $\{ \beta_{opt}/10, \zeta_{opt} \}$, and $\{ 10\beta_{opt}, \zeta_{opt} \}$, respectively). Here, $dt_0$ is tuned at each data point to control numerical errors; everywhere, it takes a value between $0.01$ and $1.0$. In comparison with the memcomputing approach, the inset shows that $T_{median}$ scales exponentially for three different physics-inspired solvers, including coherent Ising machines tiunov2019annealing, gain-dissipative dynamics kalinin2018global, and simulated bifurcation machines goto2016bifurcation. In both figures, batches of $100$ instances are run in parallel, and the batch terminates when at least half of all instances are solved. $\{\alpha/\beta, \beta, \gamma, \delta, \zeta \}_{opt} = \{0.45, 79, 0.36, 0.080, 0.063 \}$ in both (a) and (b).
• Figure 2: (a) Diagram in $\{ \beta, \zeta \}$ (in units of $\{ \beta_{opt}, \zeta_{opt} \}= \{ 79, 0.063\}$) characterizing the degree of scale-freeness of avalanche distributions, alongside (b)-(e) four of these avalanche distributions at four points in parameter space, labeled in panel (a). In diagram (a), a parameter range exists where scale-free (SF) avalanche distributions are detected with an exponent between $2$ and $3$. Of the four points in the diagram (a) whose distributions are shown, three are not SF (points b, d, and e). However, some of these distributions (at points b and e) still feature large avalanches that approach the system size. At point c, the distribution is fitted by $s^{-2.82}$ and includes a characteristic bump due to finite size effects, with finite size scaling plotted in the inset. At point d, only small avalanches are detected, as the lack of memory in the system destroys the collective behavior. 100 batches are simulated, and each batch includes 100 instances run for $T = 1000$ steps with $N=100$ (except for (c), where $N=60$ is plotted in orange, and $N=80$ is plotted in green alongside $N=100$ in blue). The time window to detect the avalanches is $\Delta_{tw} = 1.0$, $0.006$, $0.0006$, and $0.035$ for points b, c, d, and e, respectively (see the SM for more details).
• Figure 3: Diagram in $\{\beta, \zeta\}$ (in units of $\{ \beta_{opt}, \zeta_{opt} \}= \{ 79, 0.063\}$) featuring the number of time-reverse trajectories (anti-instantons) present during the DMM solution search (per batch). Anti-instantons are detected by extracting a time-ordered list of avalanches in $v_i$'s (collections of flips within a short time interval) and keeping track of when avalanches consisting of identical sets of $v_i$'s occur in immediate succession (see the SM for details). We only detect anti-instantons when $\beta$ is low. This region overlaps significantly with the dark region at the top of Fig. \ref{['fig:Fig1']}(a). 15 batches are simulated, each batch includes 100 instances each with $N = 100$, and simulations are stopped after 6000 time steps.
• Figure 4: A comparison of the time-to-solution (TTS) distributions $P(T)$ as a function of number of solution steps $T$ for different $\{ \beta, \zeta \}$. Plots (a), (b), (c), and (d) correspond to plots (b), (c), (d), and (e) in Fig. \ref{['fig:Fig2']}. The distributions in all plots are fitted to inverse Gaussians. 100 batches are simulated, and each batch includes 100 instances each with $N = 100$. According to the results of Fig. \ref{['fig:Fig1']}(a), plots (a) and (c) correspond to parameters that lead to inefficient solution search, while the parameters in (b) and (d) lead to an efficient search. For (a), the median solution step is $T_{median} \approx 10^7$. For (c), a median of instances could not be solved within $T = 10^8$ steps.