Neural Schr\"{o}dinger Equation:Physical Law as Neural Network

We show a new family of neural networks based on the Schr\"{o}dinger equation (SE-NET). In this analogy, the trainable weights of the neural networks correspond to the physical quantities of the Schr\"{o}dinger equation. These physical quantities can be trained using the complex-valued adjoint method. Since the propagation of the SE-NET can be described by the evolution of physical systems, its outputs can be computed by using a physical solver. As a demonstration, we implemented the SE-NET using the finite difference method. The trained network is transferable to actual optical systems. Based on this concept, we show a numerical demonstration of end-to-end machine learning with an optical frontend. Our results extend the application field of machine learning to hybrid physical-digital optimizations.