# Complex-valued neural networks for machine learning on non-stationary physical data

## TL;DR

This paper demonstrates that complex-valued neural networks effectively utilize phase information in non-stationary physical data, leading to improved training stability, better generalization, and parameter efficiency compared to real-valued networks.

## Contribution

It introduces the use of complex-valued deep convolutional networks for physical data, highlighting the benefits of phase information and parameter reduction over traditional real-valued models.

## Key findings

- Including phase information improves training and inference.
- Complex networks outperform larger real-valued networks.
- Parameter reduction enhances model efficiency.

## Abstract

Deep learning has become an area of interest in most scientific areas, including physical sciences. Modern networks apply real-valued transformations on the data. Particularly, convolutions in convolutional neural networks discard phase information entirely. Many deterministic signals, such as seismic data or electrical signals, contain significant information in the phase of the signal. We explore complex-valued deep convolutional networks to leverage non-linear feature maps. Seismic data commonly has a lowcut filter applied, to attenuate noise from ocean waves and similar long wavelength contributions. Discarding the phase information leads to low-frequency aliasing analogous to the Nyquist-Shannon theorem for high frequencies. In non-stationary data, the phase content can stabilize training and improve the generalizability of neural networks. While it has been shown that phase content can be restored in deep neural networks, we show how including phase information in feature maps improves both training and inference from deterministic physical data. Furthermore, we show that the reduction of parameters in a complex network outperforms larger real-valued networks.