Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2102.03547

Directed percolation and numerical stability of simulations of digital memcomputing machines

Yuan-Hang Zhang·Massimiliano Di Ventra
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.ETcond-mat.stat-mechcs.NEnlin.AO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuan-Hang Zhang, Massimiliano Di Ventra. 2021-03-31. Directed percolation and numerical stability of simulations of digital memcomputing machines. https://doi.org/10.1063/5.0045375

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond HBM-on-GPU: Thermal Design Envelope for 3D Volumetric DRAM-on-GPU Integration

The scaling of GPUs for AI and HPC workloads is increasingly constrained by the capacity, bandwidth, and thermal limits of both 2.5D HBM-GPU and direct-stacked 3D HBM-on-GPU integration. This work establishes the thermal design envelope for 3D volumetric DRAM-on-GPU integration, in which vertically oriented DRAM dies and interleaved cooling cavities reshape heat flow and memory interfacing above the GPU. Using a package-level thermal model anchored to a consistent HBM-on-GPU baseline and driven by a realistic reticle-scale non-uniform GPU power map, we quantify the key parameters governing thermal feasibility. Stack height is the dominant limiter of peak temperature, while cooling-cavity conductivity shifts the feasible region, and mold insertion and stack orientation further modulate thermal behavior. A distributed memory-controller and network-on-chip tier introduces only a moderate thermal penalty. Although die-level parallelism increases bandwidth, the reduction in simulated training time saturates once execution becomes compute-bound. These results define a bounded co-design space across bandwidth, capacity, and thermal constraints for 3D volumetric DRAM-on-GPU integration.

cs.ET↗
arXiv2026

Droop-Aware Foundation Model Power Flow

This paper develops a droop-aware extension of the GridFM power systems foundation model, embedding droop gains and frequency/voltage deadband parameters as per-bus node features to enable control-aware AC power-flow analysis. Existing power-flow datasets encode only static electrical features, conflating operating points from qualitatively different control regimes; this work resolves that gap by exposing droop and deadband parameters as structured node features, with deadband discontinuities handled through a smooth tanh approximation that preserves solver differentiability. A transformer-based graph neural network is pre-trained on masked reconstruction and fine-tuned on the resulting control-aware datasets. The framework is validated against PSCAD electromagnetic-transient simulations on a two-bus system (0.11% maximum steady-state error) and cross-validated against an independent PyPower droop solver on the IEEE 24-bus RTS. On the 24-bus system the surrogate attains R2 = 0.9996 for active generation and 0.0015 p.u. voltage-magnitude RMSE; scalability is confirmed on the IEEE 300-bus system (0.0036 p.u. RMSE, R2 = 0.9841 for voltage magnitude across 299,700 predictions). A three-mode control study further shows that the deadband widens the control-error distribution while leaving total droop compensation unchanged, establishing deadband width as an actionable node-level design feature.

cs.ET↗
arXiv2026

GNN-based Path-aware multi-view Circuit Learning for Technology Mapping

Traditional technology mapping suffers from systemic inaccuracies in delay estimation due to its reliance on abstract, technology-agnostic delay models that fail to capture the nuanced timing behavior behavior of real post-mapping circuits. To address this fundamental limitation, we introduce GPA(graph neural network (GNN)-based Path-Aware multi-view circuit learning), a novel GNN framework that learns precise, data-driven delay predictions by synergistically fusing three complementary views of circuit structure: And-Inverter Graphs (AIGs)-based functional encoding, post-mapping technology emphasizes critical timing paths. Trained exclusively on real cell delays extracted from critical paths of industrial-grade post-mapping netlists, GPA learns to classify cut delays with unprecedented accuracy, directly informing smarter mapping decisions. Evaluated on the 19 EPFL combinational benchmarks, GPA achieves 19.9%, 2.1% and 4.1% average delay reduction over the conventional heuristics methods (techmap, MCH) and the prior state-of-the-art ML-based approach SLAP, respectively-without compromising area efficiency.

cs.ET↗