Digital memcomputing machines (DMMs) are a novel, non-Turing class of machines designed to solve combinatorial optimization problems. They can be physically realized with continuous-time, non-quantum dynamical systems with memory (time non-locality), whose ordinary differential equations (ODEs) can be numerically integrated on modern computers. Solutions of many hard problems have been reported by numerically integrating the ODEs of DMMs, showing substantial advantages over state-of-the-art solvers. To investigate the reasons behind the robustness and effectiveness of this method, we employ three explicit integration schemes (forward Euler, trapezoid and Runge-Kutta 4th order) with a constant time step, to solve 3-SAT instances with planted solutions. We show that, (i) even if most of the trajectories in the phase space are destroyed by numerical noise, the solution can still be achieved; (ii) the forward Euler method, although having the largest numerical error, solves the instances in the least amount of function evaluations; and (iii) when increasing the integration time step, the system undergoes a "solvable-unsolvable transition" at a critical threshold, which needs to decay at most as a power law with the problem size, to control the numerical errors. To explain these results, we model the dynamical behavior of DMMs as directed percolation of the state trajectory in the phase space in the presence of noise. This viewpoint clarifies the reasons behind their numerical robustness and provides an analytical understanding of the unsolvable-solvable transition. These results land further support to the usefulness of DMMs in the solution of hard combinatorial optimization problems.

中文翻译：

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数字内存计算机模拟的定向渗透和数值稳定性

数字内存计算机（DMM）是一种新颖的非图灵类计算机，旨在解决组合优化问题。它们可以通过具有内存（时间非局部性）的连续时间非量子动力系统物理实现，其常微分方程（ODE）可以通过数值方式集成在现代计算机中。通过对DMM的ODE进行数值积分，已经报告了许多难题的解决方案，与传统的求解器相比，它们显示出显着的优势。为了研究此方法的鲁棒性和有效性背后的原因，我们采用了三种具有固定时间步长的显式积分方案（正向Euler，梯形和Runge-Kutta 4阶），以种植算法求解3-SAT实例。我们证明 （i）即使相空间中的大多数轨迹被数值噪声所破坏，该解决方案仍然可以实现；（ii）前向欧拉方法尽管具有最大的数值误差，但可以用最少的函数求值来求解实例；（iii）当增加积分时间步长时，系统在临界阈值处经历“可解决的不可解决的转变”，该问题最多需要作为具有问题大小的幂律而衰减，以控制数值误差。为了解释这些结果，我们将数字万用表的动态行为建模为存在噪声时在相空间中状态轨迹的定向渗透。这种观点阐明了其数值鲁棒性背后的原因，并提供了对不可解决的过渡的分析理解。