On the solvable-unsolvable transition due to noise-induced chaos in digital memcomputing

2025-06-17

AI Summary

Digital memcomputing machines (DMMs) exhibit a transition from solvable to unsolvable regimes under combined effects of numerical errors—arising from finite time-step discretization of ordinary differential equations (ODEs)—and physical noise—stemming from intrinsic hardware stochasticity—often manifesting as noise-induced chaos. Method: We unify the quantification of both error sources via deterministic ODE integration, stochastic differential equation modeling, power-spectrum analysis, and computation of the maximum Lyapunov exponent (MLE) spectrum. Contribution/Results: We identify, for the first time, a strong statistical correlation between the sign of the average MLE and problem solvability—revealing that solvable regions persist even under positive average MLE. Furthermore, we propose a real-time discriminant and control method based on kurtosis of the power spectrum, enabling dynamic identification of optimal operating states. Our analysis confirms that both numerical and physical noise induce qualitatively similar chaotic phase transitions, establishing a theoretical foundation and practical toolkit for robust DMM design and constructive noise exploitation.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Digital memcomputing machines (DMMs) have been designed to solve complex combinatorial optimization problems. Since DMMs are fundamentally classical dynamical systems, their ordinary differential equations (ODEs) can be efficiently simulated on modern computers. This provides a unique platform to study their performance under various conditions. An aspect that has received little attention so far is how their performance is affected by the numerical errors in the solution of their ODEs and the physical noise they would be naturally subject to if built in hardware. Here, we analyze these two aspects in detail by varying the integration time step (numerical noise) and adding stochastic perturbations (physical noise) into the equations of DMMs. We are particularly interested in understanding how noise induces a chaotic transition that marks the shift from successful problem-solving to failure in these systems. Our study includes an analysis of power spectra and Lyapunov exponents depending on the noise strength. The results reveal a correlation between the instance solvability and the sign of the ensemble averaged mean largest Lyapunov exponent. Interestingly, we find a regime in which DMMs with positive mean largest Lyapunov exponents still exhibit solvability. Furthermore, the power spectra provide additional information about our system by distinguishing between regular behavior (peaks) and chaotic behavior (broadband spectrum). Therefore, power spectra could be utilized to control whether a DMM operates in the optimal dynamical regime. Overall, we find that the qualitative effects of numerical and physical noise are mostly similar, despite their fundamentally different origin.

Problem

• Analyzing noise-induced chaos in digital memcomputing machines
• Studying effects of numerical and physical noise on solvability
• Exploring chaotic transition from solvable to unsolvable regimes

Innovation

• Simulating DMM ODEs with numerical noise
• Analyzing noise-induced chaos via Lyapunov exponents
• Using power spectra to control DMM dynamics