A Feedforward Neural Network Approach to Solving Systems of Linear Equations

Resumo

This paper proposes a neural network-based framework for solving systems of linear equations of the form $\mathbf{A}\mathbf{x} = \mathbf{b}$. The method reformulates the problem as a residual minimization task and employs a feedforward neural network to learn the mapping from input matrix-vector pairs to solution vectors. The network is trained using synthetic data and optimized via gradient descent using residual-based loss. Experimental results demonstrate that the model achieves high accuracy for well-conditioned systems with dimensions up to $n = 20$, producing residual errors below $10^{-4}$ in most cases. Comparative analysis against classical numerical solvers shows that while traditional methods remain superior for ill-conditioned systems, the neural approach offers notable advantages in inference speed, generalization, and suitability for parallel or real-time deployment. Limitations and future enhancements—including scalability, noise robustness, and hybridization—are also discussed.