# Local learning for stable backpropagation-free neural network training towards physical learning

arXiv: https://arxiv.org/abs/2603.24790 · HTML: https://mlsanity.com/paper/2603.24790/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 60 references that are in the map.
- First posted on arXiv 25 Mar 2026.

## Builds on (most cited first, 25 of 25)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Direct Feedback Alignment Provides Learning in Deep Neural Networks](https://mlsanity.com/paper/1609.01596/) (2016)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Gradients without Backpropagation](https://mlsanity.com/paper/2202.08587/) (2022)
- [Neuromorphic Silicon Photonic Networks](https://mlsanity.com/paper/1611.02272/) (2016)
- [Scaling Forward Gradient With Local Losses](https://mlsanity.com/paper/2210.03310/) (2022)
- [Hebbian Deep Learning Without Feedback](https://mlsanity.com/paper/2209.11883/) (2022)
- [Physical reservoir computing -- An introductory perspective](https://mlsanity.com/paper/2005.00992/) (2020)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks](https://mlsanity.com/paper/2205.08501/) (2022)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
- [Backpropagation-free Training of Deep Physical Neural Networks](https://mlsanity.com/paper/2304.11042/) (2023)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Maximum Class Separation as Inductive Bias in One Matrix](https://mlsanity.com/paper/2206.08704/) (2022)
- [Reprogrammable Electro-Optic Nonlinear Activation Functions for Optical Neural Networks](https://mlsanity.com/paper/1903.04579/) (2019)
- [Self-Contrastive Forward-Forward Algorithm](https://mlsanity.com/paper/2409.11593/) (2024)
- [Watt For What: Rethinking Deep Learning's Energy-Performance Relationship](https://mlsanity.com/paper/2310.06522/) (2023)
- [Training Convolutional Neural Networks with the Forward-Forward algorithm](https://mlsanity.com/paper/2312.14924/) (2023)
- [Matrix optimization on universal unitary photonic devices](https://mlsanity.com/paper/1808.00458/) (2018)
- [Hierarchical Architectures in Reservoir Computing Systems](https://mlsanity.com/paper/2105.06923/) (2021)
- [Parallel fault-tolerant programming of an arbitrary feedforward photonic network](https://mlsanity.com/paper/1909.06179/) (2019)
- [Function regression using the forward forward training and inferring paradigm](https://mlsanity.com/paper/2510.06762/) (2025)

## Built on by (most cited first, 1 of 1)

- [FFR: Forward-Forward Learning for Regression](https://mlsanity.com/paper/2606.03927/) (2026)

## Abstract (arXiv)

While backpropagation and automatic differentiation have driven deep learning's success, the physical limits of chip manufacturing and rising environmental costs of deep learning motivate alternative learning paradigms such as physical neural networks. However, most existing physical neural networks still rely on digital computing for training, largely because backpropagation and automatic differentiation are difficult to realize in physical systems. We introduce FFzero, a forward-only learning framework enabling stable neural network training without backpropagation or automatic differentiation. FFzero combines layer-wise local learning, prototype-based representations, and directional-derivative-based optimization through forward evaluations only. We show that local learning is effective under forward-only optimization, where backpropagation fails. FFzero generalizes to multilayer perceptron and convolutional neural networks across classification and regression. Using a simulated photonic neural network as an example, we demonstrate that FFzero provides a viable path toward backpropagation-free in-situ physical learning.