# Learning on Arbitrary Graph Topologies via Predictive Coding

arXiv: https://arxiv.org/abs/2201.13180 · HTML: https://mlsanity.com/paper/2201.13180/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 64 references that are in the map.
- First posted on arXiv 31 Jan 2022.

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

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning](https://mlsanity.com/paper/1605.08104/) (2016)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Dendritic cortical microcircuits approximate the backpropagation algorithm](https://mlsanity.com/paper/1810.11393/) (2018)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [The Neural Coding Framework for Learning Generative Models](https://mlsanity.com/paper/2012.03405/) (2020)
- [Associative Memories via Predictive Coding](https://mlsanity.com/paper/2109.08063/) (2021)
- [Artificial Intelligence and Statistics](https://mlsanity.com/paper/1712.03779/) (2017)
- [Biologically Motivated Algorithms for Propagating Local Target Representations](https://mlsanity.com/paper/1805.11703/) (2018)
- [Generalization of Equilibrium Propagation to Vector Field Dynamics](https://mlsanity.com/paper/1808.04873/) (2018)
- [Reverse Differentiation via Predictive Coding](https://mlsanity.com/paper/2103.04689/) (2021)
- [Relaxing the Constraints on Predictive Coding Models](https://mlsanity.com/paper/2010.01047/) (2020)

## Built on by (most cited first, 30 of 33)

- [Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?](https://mlsanity.com/paper/2202.09467/) (2022)
- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)
- [Benchmarking Predictive Coding Networks -- Made Simple](https://mlsanity.com/paper/2407.01163/) (2024)
- [Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning](https://mlsanity.com/paper/2206.02629/) (2022)
- [A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks](https://mlsanity.com/paper/2212.00720/) (2022)
- [A Theoretical Framework for Inference Learning](https://mlsanity.com/paper/2206.00164/) (2022)
- [A Theoretical Framework for Inference and Learning in Predictive Coding Networks](https://mlsanity.com/paper/2207.12316/) (2022)
- [Predictive Coding beyond Gaussian Distributions](https://mlsanity.com/paper/2211.03481/) (2022)
- [Sequential Memory with Temporal Predictive Coding](https://mlsanity.com/paper/2305.11982/) (2023)
- [Predictive Coding Networks and Inference Learning: Tutorial and Survey](https://mlsanity.com/paper/2407.04117/) (2024)
- [Block-local learning with probabilistic latent representations](https://mlsanity.com/paper/2305.14974/) (2023)
- [Predictive Coding beyond Correlations](https://mlsanity.com/paper/2306.15479/) (2023)
- [A Review of Neuroscience-Inspired Machine Learning](https://mlsanity.com/paper/2403.18929/) (2024)
- [Preventing Deterioration of Classification Accuracy in Predictive Coding Networks](https://mlsanity.com/paper/2208.07114/) (2022)
- [Robust Graph Representation Learning via Predictive Coding](https://mlsanity.com/paper/2212.04656/) (2022)
- [Bidirectional predictive coding](https://mlsanity.com/paper/2505.23415/) (2025)
- [Divide-and-Conquer Predictive Coding: a structured Bayesian inference algorithm](https://mlsanity.com/paper/2408.05834/) (2024)
- [Faster Predictive Coding Networks via Better Initialization](https://mlsanity.com/paper/2601.20895/) (2026)
- [pyhgf: A neural network library for predictive coding](https://mlsanity.com/paper/2410.09206/) (2024)
- [Toward Practical Equilibrium Propagation: Brain-inspired Recurrent Neural Network with Feedback Regulation and Residual Connections](https://mlsanity.com/paper/2508.11659/) (2025)
- [Deep Predictive Coding with Bi-directional Propagation for Classification and Reconstruction](https://mlsanity.com/paper/2305.18472/) (2023)
- [Bridging Predictive Coding and MDL: A Two-Part Code Framework for Deep Learning](https://mlsanity.com/paper/2505.14635/) (2025)
- [Predictive Coding Graphs are a Superset of Feedforward Neural Networks](https://mlsanity.com/paper/2603.06142/) (2026)
- [Closed-form predictive coding via hierarchical Gaussian filters](https://mlsanity.com/paper/2605.20293/) (2026)
- [Augmented Lagrangian Predictive Coding](https://mlsanity.com/paper/2605.31022/) (2026)
- [Nature-Inspired Local Propagation](https://mlsanity.com/paper/2402.05959/) (2024)
- [Predictive Attractor Models](https://mlsanity.com/paper/2410.02430/) (2024)
- [Meta-Representational Predictive Coding: Neuroscience-Informed Self-Supervised Learning](https://mlsanity.com/paper/2503.21796/) (2025)
- [DMPCN: Dynamic Modulated Predictive Coding Network with Hybrid Feedback Representations](https://mlsanity.com/paper/2504.14665/) (2025)
- [Learning Long-Range Dependencies with Temporal Predictive Coding](https://mlsanity.com/paper/2602.18131/) (2026)

## Abstract (arXiv)

Training with backpropagation (BP) in standard deep learning consists of two main steps: a forward pass that maps a data point to its prediction, and a backward pass that propagates the error of this prediction back through the network. This process is highly effective when the goal is to minimize a specific objective function. However, it does not allow training on networks with cyclic or backward connections. This is an obstacle to reaching brain-like capabilities, as the highly complex heterarchical structure of the neural connections in the neocortex are potentially fundamental for its effectiveness. In this paper, we show how predictive coding (PC), a theory of information processing in the cortex, can be used to perform inference and learning on arbitrary graph topologies. We experimentally show how this formulation, called PC graphs, can be used to flexibly perform different tasks with the same network by simply stimulating specific neurons, and investigate how the topology of the graph influences the final performance. We conclude by comparing against simple baselines trained~with~BP.