# Learning Interpolations between Boltzmann Densities

arXiv: https://arxiv.org/abs/2301.07388 · HTML: https://mlsanity.com/paper/2301.07388/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 18 Jan 2023.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Learning Lattice Quantum Field Theories with Equivariant Continuous Flows](https://mlsanity.com/paper/2207.00283/) (2022)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions](https://mlsanity.com/paper/2207.08945/) (2022)
- [Flow-based sampling for fermionic lattice field theories](https://mlsanity.com/paper/2106.05934/) (2021)
- [Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flows](https://mlsanity.com/paper/2110.02673/) (2021)
- [Gradients should stay on Path: Better Estimators of the Reverse- and Forward KL Divergence for Normalizing Flows](https://mlsanity.com/paper/2207.08219/) (2022)
- [Flow-based sampling in the lattice Schwinger model at criticality](https://mlsanity.com/paper/2202.11712/) (2022)
- [Flowification: Everything is a Normalizing Flow](https://mlsanity.com/paper/2205.15209/) (2022)
- [Integrable Nonparametric Flows](https://mlsanity.com/paper/2012.02035/) (2020)
- [Deformations of Boltzmann Distributions](https://mlsanity.com/paper/2210.13772/) (2022)

## Built on by (most cited first, 30 of 43)

- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [NETS: A Non-Equilibrium Transport Sampler](https://mlsanity.com/paper/2410.02711/) (2024)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Protein Conformation Generation via Force-Guided SE(3) Diffusion Models](https://mlsanity.com/paper/2403.14088/) (2024)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [LEAPS: A discrete neural sampler via locally equivariant networks](https://mlsanity.com/paper/2502.10843/) (2025)
- [No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers](https://mlsanity.com/paper/2502.06685/) (2025)
- [MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control](https://mlsanity.com/paper/2508.10684/) (2025)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training](https://mlsanity.com/paper/2501.06148/) (2025)
- [Accelerated Parallel Tempering via Neural Transports](https://mlsanity.com/paper/2502.10328/) (2025)
- [FEAT: Free energy Estimators with Adaptive Transport](https://mlsanity.com/paper/2504.11516/) (2025)
- [Neural Thermodynamic Integration: Free Energies from Energy-based Diffusion Models](https://mlsanity.com/paper/2406.02313/) (2024)
- [Discrete Neural Flow Samplers with Locally Equivariant Transformer](https://mlsanity.com/paper/2505.17741/) (2025)
- [Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond](https://mlsanity.com/paper/2502.04575/) (2025)
- [CREPE: Controlling Diffusion with Replica Exchange](https://mlsanity.com/paper/2509.23265/) (2025)
- [Improving the evaluation of samplers on multi-modal targets](https://mlsanity.com/paper/2504.08916/) (2025)
- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Solvation Free Energies from Neural Thermodynamic Integration](https://mlsanity.com/paper/2410.15815/) (2024)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)
- [Diffusion-PINN Sampler](https://mlsanity.com/paper/2410.15336/) (2024)
- [Telegrapher's Generative Model via Kac Flows](https://mlsanity.com/paper/2506.20641/) (2025)
- [Zero Shot Molecular Generation via Similarity Kernels](https://mlsanity.com/paper/2402.08708/) (2024)
- [Force-Guided Bridge Matching for Full-Atom Time-Coarsened Dynamics of Peptides](https://mlsanity.com/paper/2408.15126/) (2024)
- [Mitigating mode collapse in normalizing flows by annealing with an adaptive schedule: Application to parameter estimation](https://mlsanity.com/paper/2505.03652/) (2025)
- [Adaptive Destruction Processes for Diffusion Samplers](https://mlsanity.com/paper/2506.01541/) (2025)
- [Reinforced sequential Monte Carlo for amortised sampling](https://mlsanity.com/paper/2510.11711/) (2025)
- [Neural Flow Samplers with Shortcut Models](https://mlsanity.com/paper/2502.07337/) (2025)
- [Score-based deterministic density sampling](https://mlsanity.com/paper/2504.18130/) (2025)
- [Energy-based generator matching: A neural sampler for general state space](https://mlsanity.com/paper/2505.19646/) (2025)

## Abstract (arXiv)

We introduce a training objective for continuous normalizing flows that can be used in the absence of samples but in the presence of an energy function. Our method relies on either a prescribed or a learnt interpolation ft of energy functions between the target energy f1 and the energy function of a generalized Gaussian f0(x) = ||x/σ||pp. The interpolation of energy functions induces an interpolation of Boltzmann densities pt propto e-ft and we aim to find a time-dependent vector field Vt that transports samples along the family pt of densities. The condition of transporting samples along the family pt is equivalent to satisfying the continuity equation with Vt and pt = Zt-1e-ft. Consequently, we optimize Vt and ft to satisfy this partial differential equation. We experimentally compare the proposed training objective to the reverse KL-divergence on Gaussian mixtures and on the Boltzmann density of a quantum mechanical particle in a double-well potential.