ON-OFF Neuromorphic ISING Machines using Fowler-Nordheim Annealers

TL;DR

NeuroSA introduces a neuromorphic Ising machine that is functionally isomorphic to an optimal simulated annealing engine, using ON-OFF integrate-and-fire neurons paired with a Fowler-Nordheim annealer to implement the $O(1/\log)$ cooling schedule. This design yields asymptotic convergence to the Ising ground state while exploiting parallelism and noise in neuromorphic hardware, demonstrated on MAX-CUT and MIS benchmarks with distributions closely tracking or surpassing state-of-the-art solutions. The approach shows robust performance across graph scales, benefits from parallel execution on SpiNNaker2, and remains tunable via the FN dynamics and annealing parameters, with practical considerations for precision, routing, and energy efficiency. Overall, NeuroSA offers a scalable, hardware-friendly pathway to high-quality combinatorial optimization near the ground state, leveraging neuromorphic platforms to accelerate SA-like dynamics and explore novel solutions as run-time grows.

Abstract

We introduce NeuroSA, a neuromorphic architecture specifically designed to ensure asymptotic convergence to the ground state of an Ising problem using a Fowler-Nordheim quantum mechanical tunneling based threshold-annealing process. The core component of NeuroSA consists of a pair of asynchronous ON-OFF neurons, which effectively map classical simulated annealing dynamics onto a network of integrate-and-fire neurons. The threshold of each ON-OFF neuron pair is adaptively adjusted by an FN annealer and the resulting spiking dynamics replicates the optimal escape mechanism and convergence of SA, particularly at low-temperatures. To validate the effectiveness of our neuromorphic Ising machine, we systematically solved benchmark combinatorial optimization problems such as MAX-CUT and Max Independent Set. Across multiple runs, NeuroSA consistently generates distribution of solutions that are concentrated around the state-of-the-art results (within 99%) or surpass the current state-of-the-art solutions for Max Independent Set benchmarks. Furthermore, NeuroSA is able to achieve these superior distributions without any graph-specific hyperparameter tuning. For practical illustration, we present results from an implementation of NeuroSA on the SpiNNaker2 platform, highlighting the feasibility of mapping our proposed architecture onto a standard neuromorphic accelerator platform.

Figures (10)

• Figure 1: NeuroSA motivation for mapping of optimal simulated annealing into a neuromorphic architecture: (a) Illustration of the distribution of solutions generated by optimal and non-optimal Ising machines for different COP complexity: Low(L), Medium(M), and High(H). An ideal Ising/QUBO solver produces distribution of solutions that is concentrated near the SOTA and has the potential to produce novel, previously unknown solution that is closer to the Ising ground state; A MAX-CUT problem defined over a (b) graph with weights $Q_{ij}$ which is decomposed into (c) pairs of ON-OFF neurons by NeuroSA. (d) Each ON-OFF integrate-and-fire neurons are coupled to each other by an excitatory synapse with weight $A$ and the pair is connected differentially to other ON-OFF neuron pairs through the synaptic weights $Q_{ij},-Q_{ij}$. The thresholds for both ON-OFF neurons are dynamically adjusted by an (e) FN annealer which comprises an FN integrator, an exponentially-distributed noise source $\mathcal{N}_n^E$ and a Bernoulli noise source $\mathcal{N}_n^B$; Illustration of NeuroSA dynamics for a MAX-CUT graph with 10 vertices connected by a weight matrix $\mathbf{Q}$ shown in (f); (g) Evolution of the distance between the solutions generated by NeuroSA to the two known ground state solutions at a given time-instant which highlights the escape mechanisms in the high- and low-temperature regimes; (h) Raster plot of aggregated spiking activity generated by the ON and OFF neuron pairs, and (i) visualization of the NeuroSA trapping and escape dynamics using a Principle Component Analysis (PCA)-based projection of the network spiking activity estimated within a moving time-window.
• Figure 2: NeuroSA dynamics for the $G15$ MAX-CUT graph with 800 vertices and 4661 edges: (a) convergence plot showing steady increase in the solution quality with the inset showing fluctuations near $3050$ cuts which is the current SOTA for this graph; (b) dynamics of the firing threshold with inset showing sparse but large fluctuations that trigger escape mechanisms; (c) plot showing the number of active neurons decaying following $\sim\frac{1}{\log t}$ without the contribution of the Bernoulli r.v. $\mathcal{N}^B$; (d) PCA trajectory of the NeuroSA dynamics where the initial (high temperature) regime follows a path defined by the network gradient and the trajectory near convergence (or low-temperature path) exhibits expanding exploration of the solution space; and (e) distribution of the $G15$ solutions obtained for different annealing schedules ($e^{-t},(\log t)^{-1}, t^{-1}$) and noise statistics (exponential - denoted by $\mathcal{N}^E$, Gaussian - denoted by $\mathcal{N}^G$ and Uniform - denoted by $\mathcal{N}^U$)
• Figure 3: NeuroSA results for MAX-CUT and MIS benchmarks. (a)-(d) Empirical probability density functions (pdf) of the solutions on Gset benchmarks medium2019BenchmarkingMAXCUT. The solutions fall within the interval $(0.989, 1.00)$ of SOTA, indicated by the red and blue dotted lines. The results are ordered with ascending complexity: Low(L), Medium-Low(M-L), Medium(M), Medium-High(M-H), High(H). Complexity metrics: (a) the number of graph vertices, L={$800, 1,000$}, M={$2,000, 3,000$} and H={$5,000, 10,000$}; (b) the average fan-out, where L=$2.0$, M-L=$6.0$, M-H=$10.0$, H=$24.0$; (c) the graph entropy, where L=[0, 2), M=(2, 4) and H=(4, 5); (d) the graph transitivity, where L=[0, 0.001], M-L=(0.001, 0.05], M-H=(0.05, 0.14), H=[0.14, 0.16). (e) Parallel search comprising of 5 NeuroSA instances yields more consistent results than a single search running for 5$\times$ the duration of the parallel search. (f) Instantaneous time per unit gain in solution for $3$ different Gset benchmarks approaches an exponential/sub-exponential run-time. (g) Results from NeuroSA implemented on the SpiNNaker2 neuromorphic platform, where error bars plots the SD. (h) The energy-to-solution results on both CPU and SpiNNaker2 platforms, where the error bar plots the maximum and minimum energy dissipation across runs.
• Figure 4: NeuroSA results for MIS benchmarks showing that all solutions fall within the interval $(0.944, 1.11)$ of the SOTA (marked by dotted lines). Solutions are ordered according to complexity metrics: (a) number of graph vertices, with L={$10, 25, 50$}, M={$100, 250, 500$} and H={$1,000, 2,500, 5,000$}; and (b) graph density, with L=$0.01$, M-L=$0.05$, M-H=$0.1$, H=$0.25$.
• Figure S1: (a) Testing board used for the NeuroSA experiments, (b) highlighting its single SpiNNaker2 chip (c) and its internal topology.