# Training thermodynamic computers by gradient descent

arXiv: https://arxiv.org/abs/2509.15324 · HTML: https://mlsanity.com/paper/2509.15324/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Thermodynamic Computing System for AI Applications](https://mlsanity.com/paper/2312.04836/) (2023)
- [Thermodynamic Computing](https://mlsanity.com/paper/1911.01968/) (2019)
- [Thermodynamic Neural Network](https://mlsanity.com/paper/1906.01678/) (2019)
- [Solving the compute crisis with physics-based ASICs](https://mlsanity.com/paper/2507.10463/) (2025)
- [Thermodynamic Computing via Autonomous Quantum Thermal Machines](https://mlsanity.com/paper/2308.15905/) (2023)
- [Generative thermodynamic computing](https://mlsanity.com/paper/2506.15121/) (2025)
- [Nonlinear thermodynamic computing out of equilibrium](https://mlsanity.com/paper/2412.17183/) (2024)
- [Increasing the clock speed of a thermodynamic computer by adding noise](https://mlsanity.com/paper/2410.12211/) (2024)

## 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)
- [Remapping and navigation of an embedding space via error minimization: a fundamental organizational principle of cognition in natural and artificial systems](https://mlsanity.com/paper/2601.14096/) (2026)
- [Optimal Computation from Fluctuation Responses](https://mlsanity.com/paper/2510.03900/) (2025)
- [Online local learning for generative thermodynamic computing](https://mlsanity.com/paper/2609.15439/) (2026)

## Abstract (arXiv)

We show how to adjust the parameters of a thermodynamic computer by gradient descent in order to perform a desired computation at a specified observation time. Within a digital simulation of a thermodynamic computer, training proceeds by maximizing the probability with which the computer would generate an idealized dynamical trajectory. The idealized trajectory is designed to reproduce the activations of a neural network trained to perform the desired computation. This teacher-student scheme results in a thermodynamic computer whose finite-time dynamics enacts a computation analogous to that of the neural network. The parameters identified in this way can be implemented in the hardware realization of the thermodynamic computer, which will perform the desired computation automatically, driven by thermal noise. We demonstrate the method on a standard image-classification task, and estimate the thermodynamic advantage -- the ratio of energy costs of the digital and thermodynamic implementations -- to exceed seven orders of magnitude. Our results establish gradient descent as a viable training method for thermodynamic computing, enabling application of the core methodology of machine learning to this emerging field.