Accurate and scalable deep Maxwell solvers

Area of Science:

• Computational mathematics
• Applied physics
• Machine learning

Background:

• Neural networks show potential for solving partial differential equations (PDEs).
• Challenges remain in achieving high accuracy and scalability with neural network-based PDE solvers.
• Integrating neural networks with iterative algorithms is an active area of research.

Purpose of the Study:

• To develop accurate and scalable neural network surrogate solvers for partial differential equations (PDEs).
• To demonstrate the effectiveness of combining neural network surrogates with iterative algorithms.
• To enable the solution of complex, large-scale multiphysics problems.

Main Methods:

• Developed a subdomain neural operator model supporting arbitrary Robin-type boundary conditions.
• Utilized the model as a flexible preconditioner for iterative subdomain problem solving.
• Constructed global coarse spaces for accelerated, large-scale PDE solving via multilevel domain decomposition.
• Trained a single neural network on 2D Maxwell's equations for diverse problem instances.

Main Results:

• Neural network surrogates combined with iterative algorithms accurately solve PDE problems across different scales, resolutions, and boundary conditions.
• The subdomain neural operator model provides bounded accuracy for iterative subdomain solutions.
• The approach enables accelerated, large-scale PDE problem solving.
• A single trained network simulated large-scale problems with varying parameters (size, resolution, wavelength, dielectric distribution).
• Demonstrated accurate inverse design of multiwavelength nanophotonic devices.

Conclusions:

• Neural network surrogates integrated with iterative methods offer a promising path to accurate and scalable PDE solutions.
• The developed subdomain neural operator model is flexible and effective for complex boundary conditions.
• This framework facilitates efficient large-scale multiphysics simulations and inverse design applications.