# Multi-layer MIMO Relay as Deep Physical Neural Networks: Power Amplifiers as Activation Functions

arXiv: https://arxiv.org/abs/2607.18354 · HTML: https://mlsanity.com/paper/2607.18354/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 16 references that are in the map.
- First posted on arXiv 20 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Federated Learning over Wireless Fading Channels](https://mlsanity.com/paper/1907.09769/) (2019)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Over-the-Air Edge Inference via End-to-End Metasurfaces-Integrated Artificial Neural Networks](https://mlsanity.com/paper/2504.00233/) (2025)
- [Implementing Neural Networks Over-the-Air via Reconfigurable Intelligent Surfaces](https://mlsanity.com/paper/2508.01840/) (2025)
- [CNNs in the Air via Reconfigurable Intelligent Surfaces](https://mlsanity.com/paper/2510.25389/) (2025)
- [Non-Linear Relay Optimization using Deep-Learning tools](https://mlsanity.com/paper/2308.12413/) (2023)
- [Over-the-Air ODE-Inspired Neural Network for Dual Task-Oriented Semantic Communications](https://mlsanity.com/paper/2505.04970/) (2025)
- [Wireless Physical Neural Networks (WPNNs): Opportunities and Challenges](https://mlsanity.com/paper/2602.14094/) (2026)

## Abstract (arXiv)

Wireless physical neural networks (WPNNs) embed neural computation directly into analog hardware, offering lower energy consumption and latency than conventional digital implementations. In this paper, we propose a deep WPNN in which nonlinear activations are realized by a multi-hop multiple-input multiple-output (MIMO) relay network, in which each relay implements a trainable complex linear gain and bias, followed by the power amplifier's intrinsic nonlinearity acting as an activation function. The cascade of multiple relays therefore realizes an over-the-air fully connected network whose parameters can be trained end-to-end. We develop two transceiver designs for different channel state information (CSI) availability scenarios: a least squares (LS)-based scheme requiring only receiver-side CSI, and a singular-value-decomposition (SVD)-based scheme requiring both transmitter-side and receiver-side CSI. Simulation results show that the proposed architecture enables accurate over-the-air inference for image classification. In particular, the results highlight the advantage of exploiting hardware nonlinearity for enhanced inference capability.