# Online local learning for generative thermodynamic computing

arXiv: https://arxiv.org/abs/2609.15439 · HTML: https://mlsanity.com/paper/2609.15439/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 36 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Acceleration of Deep Neural Network Training with Resistive Cross-Point Devices](https://mlsanity.com/paper/1603.07341/) (2016)
- [Self-learning Machines based on Hamiltonian Echo Backpropagation](https://mlsanity.com/paper/2103.04992/) (2021)
- [Algorithm for Training Neural Networks on Resistive Device Arrays](https://mlsanity.com/paper/1909.07908/) (2019)
- [Thermodynamic Computing System for AI Applications](https://mlsanity.com/paper/2312.04836/) (2023)
- [Thermodynamic Neural Network](https://mlsanity.com/paper/1906.01678/) (2019)
- [Generative thermodynamic computing](https://mlsanity.com/paper/2506.15121/) (2025)
- [Thermodynamic Natural Gradient Descent](https://mlsanity.com/paper/2405.13817/) (2024)
- [Nonlinear thermodynamic computing out of equilibrium](https://mlsanity.com/paper/2412.17183/) (2024)
- [Local Learning Rules for Out-of-Equilibrium Physical Generative Models](https://mlsanity.com/paper/2506.19136/) (2025)
- [Training thermodynamic computers by gradient descent](https://mlsanity.com/paper/2509.15324/) (2025)
- [Neural Langevin Machine: a local asymmetric learning rule can be creative](https://mlsanity.com/paper/2506.23546/) (2025)
- [Hebbian Physics Networks: A Self-Organizing Computational Architecture Based on Local Physical Laws](https://mlsanity.com/paper/2507.00641/) (2025)

## Abstract (arXiv)

Generative thermodynamic computers turn thermal noise into structured data through Langevin dynamics. We train these systems with a local update at each integration step. The reverse-path Onsager-Machlup objective yields a coupling gradient that is a symmetric sum of local residual-state correlations. We apply this gradient immediately rather than accumulating it over a full trajectory. In digital simulations using MNIST prototypes, online and trajectory-batch training reach similar validation losses on fixed noising paths. Models trained online release less heat on average in all five independently seeded pairs, with both models' parameters held fixed during sampling. Auxiliary classifier and nearest-prototype measures change modestly, while pairwise diversity decreases. The response to noise depends strongly on where the errors enter: independent zero-mean errors in the formed updates produce little heat change over a finite range of noise amplitudes, whereas residual offset and temporal correlation have much larger effects. Storing trained couplings requires substantially less precision than resolving deterministic updates during training. Together, these results establish a local online training method and show how update timing, noise structure, and precision affect generative thermodynamic computing.