Neural Schrödinger Equation: Physical Law as Deep Neural Network

Area of Science:

• Physics
• Machine Learning
• Optical Engineering

Background:

• Neural networks are powerful tools for complex problem-solving.
• Traditional neural networks lack direct physical system integration.
• The Schrödinger equation governs quantum mechanical systems.

Purpose of the Study:

• To develop a new class of neural networks, SE-NET (Schrödinger Equation Neural Networks).
• To enable hybrid physical-digital optimization by integrating neural networks with physical systems.
• To demonstrate the application of SE-NET in optical systems, such as compact spectrometers.

Main Methods:

• Developing SE-NET where trainable weights map to physical quantities in the Schrödinger equation.
• Utilizing the complex-valued adjoint method for training.
• Implementing SE-NET with the Crank-Nicolson finite difference method on PyTorch.
• Introducing phase-only training to ensure stable training of deep SE-NET models.

Main Results:

• SE-NET performance improves with increased network width and depth.
• Deep SE-NET training initially showed instability due to gradient explosions.
• Phase-only training stabilized training for deep SE-NET models by preserving system unitarity.
• Demonstrated end-to-end machine learning with an optical frontend for a compact spectrometer.

Conclusions:

• SE-NET offers a novel framework for hybrid physical-digital optimization.
• The phase-only training method enhances the stability and applicability of deep SE-NET models.
• SE-NET extends machine learning applications to integrated physical and digital systems.