# Trainable and Dynamic Computing: Error Backpropagation through Physical Media

arXiv: https://arxiv.org/abs/1407.6637 · HTML: https://mlsanity.com/paper/1407.6637/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2014.

## Built on by (most cited first, 9 of 9)

- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Self-learning Machines based on Hamiltonian Echo Backpropagation](https://mlsanity.com/paper/2103.04992/) (2021)
- [A universal approximation theorem for nonlinear resistive networks](https://mlsanity.com/paper/2312.15063/) (2023)
- [Equilibrium Propagation for Learning in Lagrangian Dynamical Systems](https://mlsanity.com/paper/2505.07363/) (2025)
- [Training all-mechanical neural networks for task learning through in situ backpropagation](https://mlsanity.com/paper/2404.15471/) (2024)
- [Model-free Optical Processors using In Situ Reinforcement Learning with Proximal Policy Optimization](https://mlsanity.com/paper/2507.05583/) (2025)
- [Training deep physical neural networks with local physical information bottleneck](https://mlsanity.com/paper/2602.09569/) (2026)
- [Low-power analogue neural networks with trainable nonlinear connections for continuous control](https://mlsanity.com/paper/2606.23742/) (2026)

## Abstract (arXiv)

Machine learning algorithms, and more in particular neural networks, arguably experience a revolution in terms of performance. Currently, the best systems we have for speech recognition, computer vision and similar problems are based on neural networks, trained using the half-century old backpropagation algorithm. Despite the fact that neural networks are a form of analog computers, they are still implemented digitally for reasons of convenience and availability. In this paper we demonstrate how we can design physical linear dynamic systems with non-linear feedback as a generic platform for dynamic, neuro-inspired analog computing. We show that a crucial advantage of this setup is that the error backpropagation can be performed physically as well, which greatly speeds up the optimisation process. As we show in this paper, using one experimentally validated and one conceptual example, such systems may be the key to providing a relatively straightforward mechanism for constructing highly scalable, fully dynamic analog computers.