Complex-Valued Neural Networks

Complex-valued neural networks represent activations and weights as complex numbers, letting magnitude and phase interact directly rather than treating real and imaginary parts as unrelated features. Common approaches include complex convolutional and recurrent networks, unitary or phase-coherent Transformers, state-space models, and holomorphic or physics-informed networks, trained with tools such as Wirtinger derivatives. They are used for phase-rich tasks including RF and OFDM analysis, PolSAR classification, audio and EEG processing, quantum-state modeling, and prediction of physical dynamics and PDEs. Gains are most evident when phase, symmetry, or known physical structure matters, but advantages over compute- and tuning-matched real networks are not universal; robustness to phase perturbations and reliable optimization remain challenges.