# Efficient Regression-Based Training of Normalizing Flows for Boltzmann Generators arXiv: https://arxiv.org/abs/2506.01158 · HTML: https://mlsanity.com/paper/2506.01158/ ## Key facts - Cited by 8 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 48 of its 68 references that are in the map. - First posted on arXiv 1 Jun 2025. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 30 of 48) - [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [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) - [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015) - [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016) - [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018) - [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019) - [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014) - [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [WorldSimBench: Towards Video Generation Models as World Simulators](https://mlsanity.com/paper/2410.18072/) (2024) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018) - [Adversarial Diffusion Distillation](https://mlsanity.com/paper/2311.17042/) (2023) - [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019) - [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025) - [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023) - [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023) - [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024) - [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024) - [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019) - [Score identity Distillation: Exponentially Fast Distillation of Pretrained Diffusion Models for One-Step Generation](https://mlsanity.com/paper/2404.04057/) (2024) - [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023) - [Inductive Moment Matching](https://mlsanity.com/paper/2503.07565/) (2025) ## Built on by (most cited first, 8 of 8) - [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025) - [Assessing generative modeling approaches for free energy estimates in condensed matter](https://mlsanity.com/paper/2512.23930/) (2025) - [Efficient Training of Boltzmann Generators Using Off-Policy Log-Dispersion Regularization](https://mlsanity.com/paper/2602.03729/) (2026) - [Learning Structure, Energy, and Dynamics: A Survey of Artificial Intelligence for Protein Dynamics](https://mlsanity.com/paper/2604.25244/) (2026) - [Autoregressive Boltzmann Generators](https://mlsanity.com/paper/2606.27361/) (2026) - [Mode Coverage in Normalizing Flow Boltzmann Generators via Log-Ratio Variation](https://mlsanity.com/paper/2609.09473/) (2026) - [Langevin Flow Maps: Efficient Molecular Dynamics and Transition Path Sampling](https://mlsanity.com/paper/2610.05998/) (2026) - [PrT-Flow: Probability-Tractable Flow Matching for Reinforcement Learning](https://mlsanity.com/paper/2610.17667/) (2026) ## Abstract (arXiv) Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to large-scale diffusion and flow matching models. However, such modern generative models suffer from expensive inference, inhibiting their use in numerous scientific applications like Boltzmann Generators (BGs) for molecular conformations that require fast likelihood evaluation. In this paper, we revisit classical normalizing flows in the context of BGs that offer efficient sampling and likelihoods, but whose training via maximum likelihood is often unstable and computationally challenging. We propose Regression Training of Normalizing Flows (RegFlow), a novel and scalable regression-based training objective that bypasses the numerical instability and computational challenge of conventional maximum likelihood training in favour of a simple ℓ2-regression objective. Specifically, RegFlow maps prior samples under our flow to targets computed using optimal transport couplings or a pre-trained continuous normalizing flow (CNF). To enhance numerical stability, RegFlow employs effective regularization strategies such as a new forward-backward self-consistency loss that enjoys painless implementation. Empirically, we demonstrate that RegFlow unlocks a broader class of architectures that were previously intractable to train for BGs with maximum likelihood. We also show RegFlow exceeds the performance, computational cost, and stability of maximum likelihood training in equilibrium sampling in Cartesian coordinates of alanine dipeptide, tripeptide, and tetrapeptide, showcasing its potential in molecular systems.