Convergence-Enhanced and ANN-Accelerated Solvers for Absolute Value Problems

Absolute value problems of the form Ax−|x|=b, where x∈Rn is the unknown vector, b∈Rn is a given vector, and A∈Rn×n is a matrix, arise in a wide range of scientific and engineering applications. Their solution is challenging due to the non-differentiability of the absolute value operator and the possible existence of multiple solutions. Classical iterative techniques often suffer from slow convergence, strong sensitivity to the choice of initial vectors, and limited global convergence guarantees. In this study, we introduce a novel two-step iterative scheme that incorporates an adaptive initialization strategy enhanced by artificial neural networks (ANNs). The proposed method attains global linear convergence and local third-order convergence, thereby combining robustness with high accuracy. Numerical experiments on a range of benchmark problems—including cases with both unique and multiple solutions—demonstrate that the ANN-assisted initialization substantially accelerates convergence. In particular, it reduces the number of iterations, computational time, and residual errors across multiple norms, including both the Euclidean and infinity norms. These findings demonstrate that coupling a high-order two-step solver with ANN-based adaptive initialization yields a reliable and efficient framework for solving absolute value problems in both theoretical analysis and practical large-scale applications.