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

arXiv: https://arxiv.org/abs/2506.19136 · HTML: https://mlsanity.com/paper/2506.19136/

## Key facts

- Cited by 4 papers in the map (data through 14 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.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 29 of 29)

- [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)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [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)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Towards Biologically Plausible Deep Learning](https://mlsanity.com/paper/1502.04156/) (2015)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Self-learning Machines based on Hamiltonian Echo Backpropagation](https://mlsanity.com/paper/2103.04992/) (2021)
- [Training Deep Boltzmann Networks with Sparse Ising Machines](https://mlsanity.com/paper/2303.10728/) (2023)
- [Nonlinear Processing with Linear Optics](https://mlsanity.com/paper/2307.08533/) (2023)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Thermodynamic Computing System for AI Applications](https://mlsanity.com/paper/2312.04836/) (2023)
- [Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation](https://mlsanity.com/paper/2402.08579/) (2024)
- [Frequency propagation: Multi-mechanism learning in nonlinear physical networks](https://mlsanity.com/paper/2208.08862/) (2022)
- [CMOS + stochastic nanomagnets: heterogeneous computers for probabilistic inference and learning](https://mlsanity.com/paper/2304.05949/) (2023)
- [Optical Transformers](https://mlsanity.com/paper/2302.10360/) (2023)
- [Learning long range dependencies through time reversal symmetry breaking](https://mlsanity.com/paper/2506.05259/) (2025)
- [Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian Echo Learning](https://mlsanity.com/paper/2506.06248/) (2025)
- [Optical Diffusion Models for Image Generation](https://mlsanity.com/paper/2407.10897/) (2024)
- [Memristive Linear Algebra](https://mlsanity.com/paper/2407.20539/) (2024)
- [Thermodynamic Algorithms for Quadratic Programming](https://mlsanity.com/paper/2411.14224/) (2024)
- [Multimodal oscillator networks learn to solve a classification problem](https://mlsanity.com/paper/2502.12020/) (2025)
- [Nonlinear Computation with Linear Optics via Source-Position Encoding](https://mlsanity.com/paper/2504.20401/) (2025)

## Built on by (most cited first, 4 of 4)

- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [Unifying Physical Backpropagation](https://mlsanity.com/paper/2608.11585/) (2026)
- [Online local learning for generative thermodynamic computing](https://mlsanity.com/paper/2609.15439/) (2026)
- [PotentialEP: Backprop-Free Training of CIFAR-Scale Generative Models via Equilibrium Propagation](https://mlsanity.com/paper/2610.14409/) (2026)

## 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.