# BoltzNCE: Learning Likelihoods for Boltzmann Generation with Stochastic Interpolants and Noise Contrastive Estimation

arXiv: https://arxiv.org/abs/2507.00846 · HTML: https://mlsanity.com/paper/2507.00846/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 79 references that are in the map.
- First posted on arXiv 1 Jul 2025.

## Builds on (most cited first, 30 of 47)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [GraphFormers: GNN-nested Transformers for Representation Learning on Textual Graph](https://mlsanity.com/paper/2105.02605/) (2021)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [AlphaFold Meets Flow Matching for Generating Protein Ensembles](https://mlsanity.com/paper/2402.04845/) (2024)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics](https://mlsanity.com/paper/2302.01170/) (2023)
- [Density Ratio Estimation via Infinitesimal Classification](https://mlsanity.com/paper/2111.11010/) (2021)
- [Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation](https://mlsanity.com/paper/2404.19739/) (2024)
- [Equivariant Graph Neural Networks for 3D Macromolecular Structure](https://mlsanity.com/paper/2106.03843/) (2021)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)

## Built on by (most cited first, 7 of 7)

- [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025)
- [Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differences](https://mlsanity.com/paper/2605.26850/) (2026)
- [Few-Step Boltzmann Generators via Scalable Likelihood Flow Maps](https://mlsanity.com/paper/2606.29110/) (2026)
- [Autoregressive Boltzmann Generators](https://mlsanity.com/paper/2606.27361/) (2026)
- [Scalable Inference-Time Annealing with Surrogate Likelihood Estimators](https://mlsanity.com/paper/2605.31498/) (2026)
- [Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling](https://mlsanity.com/paper/2607.15682/) (2026)
- [Manifold Drift in Flow Preference Optimization: A Root Cause of Reward Hacking](https://mlsanity.com/paper/2608.20011/) (2026)

## Abstract (arXiv)

Efficient sampling from the Boltzmann distribution given its energy function is a key challenge for modeling complex physical systems such as molecules. Boltzmann Generators address this problem by leveraging continuous normalizing flows to transform a simple prior into a distribution that can be reweighted to match the target using sample likelihoods. Despite the elegance of this approach, obtaining these likelihoods requires computing costly Jacobians during integration, which is impractical for large molecular systems. To overcome this difficulty, we train an energy-based model (EBM) to approximate likelihoods using both noise contrastive estimation (NCE) and score matching, which we show outperforms the use of either objective in isolation. On 2d synthetic systems where failure can be easily visualized, NCE improves mode weighting relative to score matching alone. On alanine dipeptide, our method yields free energy profiles and energy distributions that closely match those obtained using exact likelihoods while achieving 100× faster inference. By training on multiple dipeptide systems, we show that our approach also exhibits effective transfer learning, generalizing to new systems at inference time and achieving at least a 6× speedup over standard MD. While many recent efforts in generative modeling have prioritized models with fast sampling, our work demonstrates the design of models with accelerated likelihoods, enabling the application of reweighting schemes that ensure unbiased Boltzmann statistics at scale. Our code is available at https://github.com/RishalAggarwal/BoltzNCE.