  Deep neural networks (DNNs) have achieved exceptional performance across
various fields by learning complex nonlinear mappings from large-scale
datasets. However, they encounter challenges such as high computational costs
and limited interpretability. To address these issues, hybrid approaches that
integrate physics with AI are gaining interest. This paper introduces a novel
physics-based AI model called the "Nonlinear Schr\"odinger Network", which
treats the Nonlinear Schr\"odinger Equation (NLSE) as a general-purpose
trainable model for learning complex patterns including nonlinear mappings and
memory effects from data. Existing physics-informed machine learning methods
use neural networks to approximate the solutions of partial differential
equations (PDEs). In contrast, our approach directly treats the PDE as a
trainable model to obtain general nonlinear mappings that would otherwise
require neural networks. As a type of physics-AI symbiosis, it offers a more
interpretable and parameter-efficient alternative to traditional black-box
neural networks, achieving comparable or better accuracy in some time series
classification tasks while significantly reducing the number of required
parameters. Notably, the trained Nonlinear Schr\"odinger Network is
interpretable, with all parameters having physical meanings as properties of a
virtual physical system that transforms the data to a more separable space.
This interpretability allows for insight into the underlying dynamics of the
data transformation process. Applications to time series forecasting have also
been explored. While our current implementation utilizes the NLSE, the proposed
method of using physics equations as trainable models to learn nonlinear
mappings from data is not limited to the NLSE and may be extended to other
master equations of physics.
