NEURAL NETWORK–BASED SOLVERS FOR COMPLEX-VALUED MATRIX PROBLEMS: A DATA-DRIVEN COMPUTATIONAL FRAMEWORK

Abstract

In applied math, physics, signal processing, electromagnetics, quantum mechanics, and control systems, problems with complex-valued matrices come up all the time. Straight numerical solvers for complex linear systems and matrix operations such as inversion, eigenvalue computation, and factorization frequently encounter significant computational expenses, susceptibility to ill-conditioning, and scalability constraints for extensive or real-time applications. This paper presents a neural network-based framework for addressing complex-valued matrix problems, wherein the real and imaginary components are concurrently learned through a structured learning architecture. The proposed ,method obtains stable and efficient ways for converting matrix inputs into desired solutions by applying transforming intricate matrix into real-valued representations and integrating algebraic constraints directly into the loss function. The framework use to solve a lot of different problems, like finding solutions to complex linear systems, getting close to matrix inversion, and getting close to complex operators. Numerical experiments show that the neural solver is just as accurate as traditional numerical methods, but it is better at handling noise, adapting to ill-conditioned matrices, and being more efficient when solving problems multiple times. The suggested method is a good option for data-driven numerical linear algebra in domains with complex values.