A Boltzmann machine whose effective temperature can be dynamically cooled provides a stochastic neural network realization of simulated annealing, which is an important metaheuristic for solving combinatorial or global optimization problems with broad applications in machine intelligence and operations research. However, the hardware realization of the Boltzmann stochastic element with cooling capability has never been achieved within an individual semiconductor device. Here we demonstrate a new memristive device concept based on two-dimensional material heterostructures that enables this critical stochastic element in a Boltzmann machine. The dynamic cooling effect in simulated annealing can be emulated in this multi-terminal memristive device through electrostatic bias with sigmoidal thresholding distributions. We also show that a machine-learning-based method is efficient for device-circuit co-design of the Boltzmann machine based on the stochastic memristor devices in simulated annealing. The experimental demonstrations of the tunable stochastic memristors combined with the machine-learning-based device-circuit co-optimization approach for stochastic-memristor-based neural-network circuits chart a pathway for the efficient hardware realization of stochastic neural networks with applications in a broad range of electronics and computing disciplines.
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