# Variational Neural Annealing

arXiv: https://arxiv.org/abs/2101.10154 · HTML: https://mlsanity.com/paper/2101.10154/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 106 references that are in the map.
- First posted on arXiv 25 Jan 2021.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Minimizing Finite Sums with the Stochastic Average Gradient](https://mlsanity.com/paper/1309.2388/) (2013)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Dilated Recurrent Neural Networks](https://mlsanity.com/paper/1710.02224/) (2017)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines](https://mlsanity.com/paper/1610.02746/) (2016)
- [Learning Thermodynamics with Boltzmann Machines](https://mlsanity.com/paper/1606.02718/) (2016)
- [Controlled Online Optimization Learning (COOL): Finding the ground state of spin Hamiltonians with reinforcement learning](https://mlsanity.com/paper/2003.00011/) (2020)
- [Automatic Differentiable Monte Carlo: Theory and Application](https://mlsanity.com/paper/1911.09117/) (2019)
- [Identifying Quantum Phase Transitions with Adversarial Neural Networks](https://mlsanity.com/paper/1710.08382/) (2017)
- [Natural evolution strategies and variational Monte Carlo](https://mlsanity.com/paper/2005.04447/) (2020)

## Built on by (most cited first, 30 of 47)

- [Combinatorial Optimization with Physics-Inspired Graph Neural Networks](https://mlsanity.com/paper/2107.01188/) (2021)
- [A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization](https://mlsanity.com/paper/2406.01661/) (2024)
- [Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics](https://mlsanity.com/paper/2502.08696/) (2025)
- [Variational Annealing on Graphs for Combinatorial Optimization](https://mlsanity.com/paper/2311.14156/) (2023)
- [From Tensor Network Quantum States to Tensorial Recurrent Neural Networks](https://mlsanity.com/paper/2206.12363/) (2022)
- [Neural-network solutions to stochastic reaction networks](https://mlsanity.com/paper/2210.01169/) (2022)
- [Quantum HyperNetworks: Training Binary Neural Networks in Quantum Superposition](https://mlsanity.com/paper/2301.08292/) (2023)
- [The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems](https://mlsanity.com/paper/2302.08347/) (2023)
- [Recurrent neural network wave functions for Rydberg atom arrays on kagome lattice](https://mlsanity.com/paper/2405.20384/) (2024)
- [Supplementing Recurrent Neural Networks with Annealing to Solve Combinatorial Optimization Problems](https://mlsanity.com/paper/2207.08189/) (2022)
- [Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy](https://mlsanity.com/paper/2207.14314/) (2022)
- [Investigating Topological Order using Recurrent Neural Networks](https://mlsanity.com/paper/2303.11207/) (2023)
- [Efficient Training of Multi-task Neural Solver for Combinatorial Optimization](https://mlsanity.com/paper/2305.06361/) (2023)
- [Message Passing Variational Autoregressive Network for Solving Intractable Ising Models](https://mlsanity.com/paper/2404.06225/) (2024)
- [A Bayesian generative neural network framework for epidemic inference problems](https://mlsanity.com/paper/2111.03383/) (2021)
- [Demonstrating Real Advantage of Machine-Learning-Enhanced Monte Carlo for Combinatorial Optimization](https://mlsanity.com/paper/2510.19544/) (2025)
- [Quantum approximate optimization via learning-based adaptive optimization](https://mlsanity.com/paper/2303.14877/) (2023)
- [Retentive Neural Quantum States: Efficient Ans"atze for Ab Initio Quantum Chemistry](https://mlsanity.com/paper/2411.03900/) (2024)
- [Distributionally Robust Free Energy Principle for Decision-Making](https://mlsanity.com/paper/2503.13223/) (2025)
- [Annealing in variational inference mitigates mode collapse: A theoretical study on Gaussian mixtures](https://mlsanity.com/paper/2602.12923/) (2026)
- [Quantum Approximate Optimization Algorithm applied to the binary perceptron](https://mlsanity.com/paper/2112.10219/) (2021)
- [Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths](https://mlsanity.com/paper/2202.02750/) (2022)
- [Toward Neural Network Simulation of Variational Quantum Algorithms](https://mlsanity.com/paper/2211.02929/) (2022)
- [Lattice Protein Folding with Variational Annealing](https://mlsanity.com/paper/2502.20632/) (2025)
- [LMask: Learn to Solve Constrained Routing Problems with Lazy Masking](https://mlsanity.com/paper/2505.17938/) (2025)
- [IsingFormer: Augmenting Parallel Tempering With Learned Proposals](https://mlsanity.com/paper/2509.23043/) (2025)
- [Diffusion-Augmented Markov Decision Processes for Maximum Entropy Reinforcement Learning](https://mlsanity.com/paper/2512.02019/) (2025)
- [A Generative Neural Annealer for Black-Box Combinatorial Optimization](https://mlsanity.com/paper/2505.09742/) (2025)
- [Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo](https://mlsanity.com/paper/2605.13807/) (2026)
- [Scalable Physics-Inspired Transformers for Spin Glasses](https://mlsanity.com/paper/2606.22984/) (2026)

## Abstract (arXiv)

Many important challenges in science and technology can be cast as optimization problems. When viewed in a statistical physics framework, these can be tackled by simulated annealing, where a gradual cooling procedure helps search for groundstate solutions of a target Hamiltonian. While powerful, simulated annealing is known to have prohibitively slow sampling dynamics when the optimization landscape is rough or glassy. Here we show that by generalizing the target distribution with a parameterized model, an analogous annealing framework based on the variational principle can be used to search for groundstate solutions. Modern autoregressive models such as recurrent neural networks provide ideal parameterizations since they can be exactly sampled without slow dynamics even when the model encodes a rough landscape. We implement this procedure in the classical and quantum settings on several prototypical spin glass Hamiltonians, and find that it significantly outperforms traditional simulated annealing in the asymptotic limit, illustrating the potential power of this yet unexplored route to optimization.