A feedforward neural network approach to solving systems of linear equations

Rashad Al-Jawfi

doi:10.5269/bspm.78591

Rashad Al-Jawfi prof

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.

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Referências

Strang, G. (2006). Linear Algebra and Its Applications (4th ed.). Thomson, Brooks/Cole.

Saad, Y. (2003). Iterative Methods for Sparse Linear Systems (2nd ed.). SIAM.

Benzi, M. (2002). Preconditioning techniques for large linear systems: A survey. Journal of Computational Physics, 182(2), 418–477.

Haykin, S. (2009). Neural Networks and Learning Machines (3rd ed.). Pearson Education.

LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436–444.

Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323(6088), 533–536.

Gao, X., & Wang, J. (2003). A recurrent neural network for solving linear equations. Neural Processing Letters, 17(1), 59–71.

Zhang, X., & Wang, J. (2004). Global exponential convergence of neural networks for solving linear equations. IEEE Transactions on Circuits and Systems I: Regular Papers, 51(9), 1783–1790.

Hussain, A., Zhu, Q., & Nandi, A. K. (2020). A survey on neural network-based numerical solvers for systems of equations. Neural Computing and Applications, 32(8), 4423–4444.

Trefethen, L. N., & Bau, D. (1997). Numerical Linear Algebra. SIAM.

Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM.

Nair, V., & Hinton, G. E. (2010). Rectified linear units improve restricted Boltzmann machines. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), 807–814.

Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.

Abadi, M., et al. (2016). TensorFlow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI), 265–283.

Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.