Data-Driven Neural Network Solvers for Complex-Valued Matrix Computations

Abstract

The issues of complex-valued matrices are rampant in applied mathematics, physics, signal processing,
electromagnetism, quantum mechanics, and control systems. In complex linear systems and matrix computations such
as factorization, eigenvalues and inversion, straight numerical solutions can be expensive and prone to ill-conditioning,
and the size of a problem may be limited to large scale or even real-time computing. To address complex-valued
matrix problems, this study suggests a neural network-based model whereby structured learning framework is
employed to learn complex-valued problems simultaneously between the real and imaginary components. The
proposed method will result in robust and efficient representations of complex matrices through numerical values and
algebraic constraints directly integrated into the loss, which will provide credible and efficient methods of converting
matrix-based inputs to the desired outputs. Much of the activity in the field is managed using the framework, including
solving complex linear systems, finding approximations to complex operators, and finding approximations to the
inversion of matrices. Numerical experiments show that the neural solver is as precise as traditional numerical
methods, but has fewer noise characteristics, capability to adapt to ill-conditioned matrices, and is faster to solve
problems repeatedly. The proposed approach is suitable to use in the data-driven numerical linear algebra in areas with
complex values.