mlsanityData to 7 Oct 2026

Learning Interpolations between Boltzmann Densities

arXiv 2301.07388 ↗First posted 18 Jan 2023stat.ML cs.LGcited by 42

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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.

Key facts. Cited by 42 papers in the map (data through 7 Oct 2026). Builds on 26 of its 41 references that are in the map. First posted on arXiv 18 Jan 2023.

Builds on

26 of 41 references.

References · 26

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  4. 2022Flow Matching for Generative Modelingcited by 5,426
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  6. 2018Neural Ordinary Differential Equationscited by 3,733
  7. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  8. 2015Variational Inference with Normalizing Flowscited by 2,161
  9. 2016Density estimation using Real NVPcited by 2,029
  10. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  11. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 248
  12. 2020Stochastic Normalizing Flowscited by 156
14 more
  1. 2022Flow Annealed Importance Sampling Bootstrapcited by 124
  2. 2020Sampling using SU(N) gauge equivariant flowscited by 83
  3. 2020SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flowscited by 71
  4. 2020Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Modelscited by 66
  5. 2022Learning Lattice Quantum Field Theories with Equivariant Continuous Flowscited by 37
  6. 2021Introduction to Normalizing Flows for Lattice Field Theorycited by 35
  7. 2022Gauge-equivariant flow models for sampling in lattice field theories with pseudofermionscited by 30
  8. 2021Flow-based sampling for fermionic lattice field theoriescited by 27
  9. 2021Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flowscited by 24
  10. 2022Gradients should stay on Path: Better Estimators of the Reverse- and Forward KL Divergence for Normalizing Flowscited by 22
  11. 2022Flow-based sampling in the lattice Schwinger model at criticalitycited by 20
  12. 2022Flowification: Everything is a Normalizing Flowcited by 5
  13. 2020Integrable Nonparametric Flowscited by 3
  14. 2022Deformations of Boltzmann Distributionscited by 1

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Citing papers per year: 2024: 13; 2025: 23; 2026: 6.

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  10. 2025Telegrapher's Generative Model via Kac Flowscited by 7
  11. 2024Zero Shot Molecular Generation via Similarity Kernelscited by 6
  12. 2024Force-Guided Bridge Matching for Full-Atom Time-Coarsened Dynamics of Peptidescited by 6
  13. 2025Adaptive Destruction Processes for Diffusion Samplerscited by 6
  14. 2025Neural Flow Samplers with Shortcut Modelscited by 5
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  18. 2025Energy-based generator matching: A neural sampler for general state spacecited by 3
  19. 2025Diffusion-Augmented Markov Decision Processes for Maximum Entropy Reinforcement Learningcited by 2
  20. 2024Multi-Lattice Sampling of Quantum Field Theories via Neural Operator-based Flowscited by 1
  21. 2024Mixed Noise and Posterior Estimation with Conditional DeepGEMcited by 1
  22. 2025Learning Paths for Dynamic Measure Transport: A Control Perspectivecited by 1
  23. 2025Assessing generative modeling approaches for free energy estimates in condensed mattercited by 1
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