Title: BRAIN: Boltzmann Reinforcement For Analog Ising Networks
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Reinforcement learning; Ising systems; MCMC

Abstract: Analog Ising machines represent a transformative paradigm for combinatorial optimization, exploiting physical dynamics to achieve high speed, energy efficient operations over conventional digital electronics. However, existing optimization algorithms fail to harness these platforms' massive parallelism while tackling the inherent measurement noise for Analog Ising machines. We introduce BRAIN (Boltzmann Reinforcement for Analog Ising Networks), transforming the traditional sampling-based optimization to distribution learning framework. The Boltzmann distribution provides the fundamental link between statistical physics and Ising-type combinatorial optimization, establishing the theoretical framework that enables physical systems to solve NP-hard problems. Unlike Monte Carlo Markov Chain (MCMC) methods that sample states from Boltzmann distributions, BRAIN directly learns the Boltzmann distribution through variational reinforcement learning. This fundamental transformation makes the algorithm inherently resilient to the Gaussian measurement noise intrinsic to analog Ising systems. Our approach employs policy gradients to optimize a parametric state generator, naturally aggregating information across multiple noisy measurements without requiring precise energy differences. We benchmark BRAIN against MCMC methods across diverse combinatorial optimization problems, demonstrating three critical advantages. First, BRAIN generalizes across different interaction topologies, performing effectively on both Curie-Weiss and 2D nearest-neighbor Ising models. Second, it exhibits remarkable robustness under severe measurement noise up to 40\%. Third, it scales efficiently to large systems of 65,536 (N) spins, scaling as $\mathrm{O} (N^{1.55})$, with noisy energy evaluations. With realistic 3\% Gaussian noise, BRAIN maintains 98\% ground state fidelity while MCMC methods achieve only 51\% fidelity with BRAIN arriving at the MCMC solution $192\times$ faster. Beyond ground state optimization, BRAIN preserves the complete thermodynamic landscape, analyzing phase transitions and metastable states essential for robust large scale combinatorial optimization and complex many-body physics applications.