# Learning with springs and sticks

arXiv: https://arxiv.org/abs/2508.19015 · HTML: https://mlsanity.com/paper/2508.19015/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 62 references that are in the map.
- First posted on arXiv 26 Aug 2025.

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Quantifying the Carbon Emissions of Machine Learning](https://mlsanity.com/paper/1910.09700/) (2019)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [A Survey of Neuromorphic Computing and Neural Networks in Hardware](https://mlsanity.com/paper/1705.06963/) (2017)
- [LLMCarbon: Modeling the end-to-end Carbon Footprint of Large Language Models](https://mlsanity.com/paper/2309.14393/) (2023)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Neural Network Diffusion](https://mlsanity.com/paper/2402.13144/) (2024)
- [Continuous Time Analysis of Momentum Methods](https://mlsanity.com/paper/1906.04285/) (2019)
- [GenPhys: From Physical Processes to Generative Models](https://mlsanity.com/paper/2304.02637/) (2023)
- [Thermodynamic Computing System for AI Applications](https://mlsanity.com/paper/2312.04836/) (2023)
- [Thermodynamic AI and the fluctuation frontier](https://mlsanity.com/paper/2302.06584/) (2023)
- [Neural Thermodynamic Integration: Free Energies from Energy-based Diffusion Models](https://mlsanity.com/paper/2406.02313/) (2024)
- [An Artificial Spiking Quantum Neuron](https://mlsanity.com/paper/1907.06269/) (2019)
- [Fast quantum algorithm for differential equations](https://mlsanity.com/paper/2306.11802/) (2023)
- [Speed Limits for Deep Learning](https://mlsanity.com/paper/2307.14653/) (2023)

## Abstract (arXiv)

Learning is a physical process. Here, we aim to study a simple dynamical system composed of springs and sticks capable of arbitrarily approximating any continuous function. The main idea of our work is to use the sticks to mimic a piecewise-linear approximation of the given function, use the potential energy of springs to encode a desired mean squared error loss function, and converge to a minimum-energy configuration via dissipation. We apply the proposed simulation system to regression tasks and show that its performance is comparable to that of multi-layer perceptrons. In addition, we study the thermodynamic properties of the system and find a relation between the free energy change of the system and its ability to learn an underlying data distribution. We empirically find a thermodynamic learning barrier for the system caused by the fluctuations of the environment, whereby the system cannot learn if its change in free energy hits such a barrier. We believe this simple model can help us better understand learning systems from a physical point of view.