# Analog-Friendly Predictive Coding without Activation Derivatives

arXiv: https://arxiv.org/abs/2609.32350 · HTML: https://mlsanity.com/paper/2609.32350/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 33 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 28 of 28)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Power Hungry Processing: Watts Driving the Cost of AI Deployment?](https://mlsanity.com/paper/2311.16863/) (2023)
- [The Hardware Lottery](https://mlsanity.com/paper/2009.06489/) (2020)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [The free energy principle for action and perception: A mathematical review](https://mlsanity.com/paper/1705.09156/) (2017)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Learning on Arbitrary Graph Topologies via Predictive Coding](https://mlsanity.com/paper/2201.13180/) (2022)
- [Benchmarking Predictive Coding Networks -- Made Simple](https://mlsanity.com/paper/2407.01163/) (2024)
- [LocoProp: Enhancing BackProp via Local Loss Optimization](https://mlsanity.com/paper/2106.06199/) (2021)
- [μPC: Scaling Predictive Coding to 100+ Layer Networks](https://mlsanity.com/paper/2505.13124/) (2025)
- [Relaxing the Constraints on Predictive Coding Models](https://mlsanity.com/paper/2010.01047/) (2020)
- [Lifted Bregman Training of Neural Networks](https://mlsanity.com/paper/2208.08772/) (2022)
- [Only Strict Saddles in the Energy Landscape of Predictive Coding Networks?](https://mlsanity.com/paper/2408.11979/) (2024)
- [Solving the compute crisis with physics-based ASICs](https://mlsanity.com/paper/2507.10463/) (2025)
- [Towards the Training of Deeper Predictive Coding Neural Networks](https://mlsanity.com/paper/2506.23800/) (2025)
- [ePC: Fast and Deep Predictive Coding in Digital Simulation](https://mlsanity.com/paper/2505.20137/) (2025)
- [Optoelectronic Intelligence](https://mlsanity.com/paper/2010.08690/) (2020)
- [Generalised Perceptron Learning](https://mlsanity.com/paper/2012.03642/) (2020)
- [Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation](https://mlsanity.com/paper/2411.02001/) (2024)
- [Training a Predictive Coding Network on ImageNet using Equilibrium Propagation](https://mlsanity.com/paper/2606.03584/) (2026)
- [On the Infinite Width and Depth Limits of Predictive Coding Networks](https://mlsanity.com/paper/2602.07697/) (2026)
- [Generative Models on Analog Hardware with Dynamics](https://mlsanity.com/paper/2606.27294/) (2026)
- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [Physical Foundation Models: Fixed hardware implementations of large-scale neural networks](https://mlsanity.com/paper/2604.27911/) (2026)
- [Augmented Lagrangian Predictive Coding](https://mlsanity.com/paper/2605.31022/) (2026)

## Abstract (arXiv)

Predictive coding (PC) is a local, energy-based alternative to backpropagation (BP) whose iterative inference dynamics make it attractive for implementation on analog hardware. However, standard nonlinear PC requires evaluating the derivative of the activation function during both inference and learning, which can be difficult to realise physically. Here, we introduce textitactivation-matched Bregman PC, replacing standard squared-error energies with Bregman divergences matched to the activation function. This formulation eliminates activation derivatives and, when combined with inference via mirror descent, yields local inference and learning rules requiring only weighted sums, local prediction errors, state integration, and the activation function. In digital experiments, Bregman PC performs comparably to standard PC and BP on classification and generative tasks, while preserving characteristic learning dynamics of PC and its convergence to BP under stable large-model parameterisations. These results provide a more analog-friendly formulation of nonlinear PC while retaining its key computational properties.