mlsanityData to 7 Oct 2026

Local Learning Rules for Out-of-Equilibrium Physical Generative Models

arXiv 2506.19136 ↗First posted 23 Jun 2025cs.LG cond-mat.mes-hall cs.ETcited by 2

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We show that the out-of-equilibrium driving protocol of score-based generative models (SGMs) can be learned via local learning rules. The gradient with respect to the parameters of the driving protocol is computed directly from force measurements or from observed system dynamics. As a demonstration, we implement an SGM in a network of driven, nonlinear, overdamped oscillators coupled to a thermal bath. We first apply it to the problem of sampling from a mixture of two Gaussians in 2D. Finally, we train an oscillator network on the MNIST dataset to generate images of handwritten digits 0 and 1.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 29 of its 75 references that are in the map. First posted on arXiv 23 Jun 2025.

Builds on

29 of 75 references.

Closest ancestors · 5

Same direction, newest first

  1. 2024Optical Diffusion Models for Image Generation15 Jul
  2. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusion14 Dec
  3. 2021Score-based Generative Modeling in Latent Space10 Jun
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  5. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 24

  1. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  2. 2022Flow Matching for Generative Modelingcited by 5,426
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2016Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagationcited by 251
  5. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 224
  6. 2024An analytic theory of creativity in convolutional diffusion modelscited by 103
  7. 2015Towards Biologically Plausible Deep Learningcited by 66
  8. 2024Training of Physical Neural Networkscited by 47
  9. 2023Training an Ising Machine with Equilibrium Propagationcited by 29
  10. 2021Self-learning Machines based on Hamiltonian Echo Backpropagationcited by 28
  11. 2023Training Deep Boltzmann Networks with Sparse Ising Machinescited by 23
  12. 2023Nonlinear Processing with Linear Opticscited by 19
12 more
  1. 2023Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterialcited by 19
  2. 2023Thermodynamic Computing System for AI Applicationscited by 16
  3. 2024Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagationcited by 16
  4. 2022Frequency propagation: Multi-mechanism learning in nonlinear physical networkscited by 11
  5. 2023Optical Transformerscited by 10
  6. 2023CMOS + stochastic nanomagnets: heterogeneous computers for probabilistic inference and learningcited by 10
  7. 2025Learning long range dependencies through time reversal symmetry breakingcited by 6
  8. 2025Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian Echo Learningcited by 5
  9. 2024Memristive Linear Algebracited by 3
  10. 2024Thermodynamic Algorithms for Quadratic Programmingcited by 2
  11. 2025Multimodal oscillator networks learn to solve a classification problemcited by 1
  12. 2025Nonlinear Computation with Linear Optics via Source-Position Encodingcited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computingcited by 1
  2. 2026Online local learning for generative thermodynamic computingcited by 0

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  3. 2023Micro-Macro Consistency in Multiscale Modeling: Score-Based Model Assisted Sampling of Fast/Slow Dynamical Systems
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  5. 2026Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling
  6. 2024An analysis of the noise schedule for score-based generative models
  7. 2025PINGS: Physics-Informed Neural Network for Fast Generative Sampling
  8. 2025Combining complex Langevin dynamics with score-based and energy-based diffusion models