# Universality of physical neural networks with multivariate nonlinearity

arXiv: https://arxiv.org/abs/2509.05420 · HTML: https://mlsanity.com/paper/2509.05420/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 6 Sep 2025.

## Builds on (most cited first, 7 of 7)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [All-Optical Machine Learning Using Diffractive Deep Neural Networks](https://mlsanity.com/paper/1804.08711/) (2018)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks](https://mlsanity.com/paper/2205.08501/) (2022)
- [The physics of optical computing](https://mlsanity.com/paper/2308.00088/) (2023)
- [Nonlinear Processing with Linear Optics](https://mlsanity.com/paper/2307.08533/) (2023)
- [Nonlinear optical encoding enabled by recurrent linear scattering](https://mlsanity.com/paper/2307.08558/) (2023)

## Abstract (arXiv)

The enormous energy demand of artificial intelligence is driving the development of alternative hardware for deep learning. Physical neural networks try to exploit physical systems to perform machine learning more efficiently. In particular, optical systems can calculate with light using negligible energy. While their computational capabilities were long limited by the linearity of optical materials, nonlinear computations have recently been demonstrated through modified input encoding. Despite this breakthrough, our inability to determine if physical neural networks can learn arbitrary relationships between data -- a key requirement for deep learning known as universality -- hinders further progress. Here we present a fundamental theorem that establishes a universality condition for physical neural networks. It provides a powerful mathematical criterion that imposes device constraints, detailing how inputs should be encoded in the tunable parameters of the physical system. Based on this result, we propose a scalable architecture using free-space optics that is provably universal and achieves high accuracy on image classification tasks. Further, by combining the theorem with temporal multiplexing, we present a route to potentially huge effective system sizes in highly practical but poorly scalable on-chip photonic devices. Our theorem and scaling methods apply beyond optical systems and inform the design of a wide class of universal, energy-efficient physical neural networks, justifying further efforts in their development.