AI-Powered Numerical Methods: Transforming Mathematical Problem-Solving

Abstract

This paper will discuss the impact of the emergence of Artificial Intelligence (AI) on the numerical methods and how it is revolutionising mathematics problem solving. But there are cases where the traditional numeric processes can be limited by complex, non-linear and large-scale problems. The paper integrates the use of AI-based processes - machine learning and neural networks - with traditional numeric processes to address the issues. The time, accuracy, rate of convergence and error were the key comparison metrics. The results show that the traditional numeric processes had an 85-percent accuracy and 15-percent error and the AI-based processes had a 93-percent accuracy and 7-percent error. The combination of the two performed best with an accuracy of 96-percent and an error of 4-percent. The AIs reduced the time by 36 and the hybrid model by 52 percent. Convergence study also demonstrated that the hybrid model required fewer iterations (10) to converge than the traditional methods (25). Similarly, other error measures (mean absolute error (0.02) and root mean square error (0.04) depicted the hybrid model's superiority. The findings show that the AI-based numerical approaches can assist to improve the accuracy, speed and capacity of mathematical calculations. The take home message of this paper is that the hybrid approach of AI and traditional numerical methods is a promising approach to solve non-traditional problems and also contribute to the field of numerical mathematics.

REFERENCES

[1] C. Chesneau and A. Artault, "On a comparative study on some trigonometric classes of distributions by the analysis of practical data sets," Journal of Nonlinear Modeling and Analysis, vol. 3, no. 2, pp. 225–262, 2021.

[2] R. Pemantle, M. C. Wilson, and S. Melczer, Analytic Combinatorics in Several Variables, vol. 212. Cambridge, U.K.: Cambridge University Press, 2024.

[3] F. Neese, "The SHARK integral generation and digestion system," Journal of Computational Chemistry, vol. 44, no. 3, pp. 381–396, 2023.

[4] T. Dash, S. Chitlangia, A. Ahuja, and A. Srinivasan, "A review of some techniques for inclusion of domain-knowledge into deep neural networks," Scientific Reports, vol. 12, no. 1, p. 1040, 2022.

[5] A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Y. Novokshenov, Painlevé Transcendents: The Riemann-Hilbert Approach, vol. 128. Providence, RI, USA: American Mathematical Society, 2023.

[6] H. Iwaniec and E. Kowalski, Analytic Number Theory, vol. 53. Providence, RI, USA: American Mathematical Society, 2021.

[7] M. Burgos, S. Bueno, O. Pérez, and J. D. Godino, "Onto-semiotic complexity of the definite integral," Journal of Research in Mathematics Education, vol. 10, no. 1, pp. 4–40, 2021.

[8] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, vol. 368. Providence, RI, USA: American Mathematical Society, 2022.

[9] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, vol. 40. Providence, RI, USA: American Mathematical Society, 2025.

[10] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, vol. 83. Providence, RI, USA: American Mathematical Society, 2022.

[11] I. Ayoob, "On the evaluation of certain unsolved definite integrals," European Journal of Pure and Applied Mathematics, vol. 18, no. 3, pp. 6575–6575, 2025.

[12] J. Terven, D. M. Cordova-Esparza, J. A. Romero-González, A. Ramírez-Pedraza, and E. A. Chavez-Urbiola, "A comprehensive survey of loss functions and metrics in deep learning," Artificial Intelligence Review, vol. 58, no. 7, p. 195, 2025.

[13] S. G. Krantz, Real Analysis and Foundations. Boca Raton, FL, USA: Chapman and Hall/CRC, 2022.

[14] J. Duoandikoetxea, Fourier Analysis, vol. 29. Providence, RI, USA: American Mathematical Society, 2024.

[15] S. T. Fife and J. D. Gossner, "Deductive qualitative analysis: Evaluating, expanding, and refining theory," International Journal of Qualitative Methods, vol. 23, p. 16094069241244856, 2024.

[16] G. H. Hardy, Divergent Series, vol. 334. Providence, RI, USA: American Mathematical Society, 2024.

[17] T. Lu and Q. Chen, "Visualization analysis of weak interactions in chemical systems," Comprehensive Computational Chemistry, vol. 2, pp. 240–264, 2024.

[18] D. F. Gray, The Observation and Analysis of Stellar Photospheres. Cambridge, U.K.: Cambridge University Press, 2021.

[19] K. Matsumoto, "On the analytic continuation of various multiple zeta-functions," in Number Theory for the Millennium II, AK Peters/CRC Press, 2024, pp. 417–440.

[20] D. A. Cox, Primes of the Form x2+ ny2: Fermat, Class Field Theory, and Complex Multiplication. with Solutions, vol. 387. Providence, RI, USA: American Mathematical Society, 2022.

[21] L. C. Evans, Measure Theory and Fine Properties of Functions. Boca Raton, FL, USA: Chapman and Hall/CRC, 2025.

[22] U. Grenander, Toeplitz Forms and Their Applications, vol. 321. Providence, RI, USA: American Mathematical Society, 2026.

[23] O. A. Malik, "Agentic AI deployment in infrastructure-limited environments: Observability gaps, failure modes, and AI governance primitives," Journal of Engineering and Computational Intelligence Review, vol. 4, no. 1, pp. 1–11, 2026.

[24] S. Ahmed and M. Asif, “Comparative analysis of attitudes toward climate change policies across urban and rural populations,” Pakistan Journal of Social Science Review, vol. 5, no. 1, pp. 747–769, 2026, doi: 10.5281/zenodo.18457821.

[25] S. Ahmed and M. Asif, “Public opinion on the effectiveness of local government anti-corruption measures: A multi-city survey analysis,” International Journal of Social Sciences Bulletin, vol. 4, no. 1, pp. 1189–1201, 2026, doi: 10.5281/zenodo.18412790.

[26] M. Asif and S. Ullah, “Determinants of support for federalism vs. centralization: A survey of public opinion in Punjab and Khyber Pakhtunkhwa (KP),” Social Science Review Archives, vol. 4, no. 1, pp. 2791–2807, 2026, doi: 10.70670/sra.v4i1.1843.

[27] M. Asif and S. Ullah, “Performance voting vs. identity voting: An analysis of electoral behaviour in Pakistani districts,” Journal of Applied Linguistics and TESOL (JALT), vol. 9, no. 1, pp. 213–226, 2026, doi: 10.63878/cjssr.v4i1.2079.

[28] M. Asif, A. Ali, and F. A. Shaheen, “Assessing the effects of artificial intelligence in revolutionizing human resource management: A systematic review,” Social Science Review Archives, vol. 3, no. 4, pp. 2887–2908, 2025, doi: 10.70670/sra.v3i3.1055.

[29] M. Asif and R. J. Asghar, “Managerial accounting as a driver of financial performance and sustainability in small and medium enterprises in Pakistan,” Center for Management Science Research, vol. 3, no. 7, pp. 150–163, 2025, doi: 10.5281/zenodo.17596478.

[30] D. Mohiuddin, “Adaptive marketing systems and consumer feedback loops: Implications for market development in emerging economies,” Journal of Business Insight and Innovation, vol. 5, no. 1, pp. 37–48, 2026.

[31] D. Mohiuddin, “HR tech adoption in digital banking: Implications for workforce development and financial sector growth in emerging economies,” Journal of Business Insight and Innovation, vol. 4, no. 2, pp. 77–90, 2025.

[32] D. Mohiuddin and D. N. Farhan, “Artificial intelligence in marketing: Ethical challenges and solutions for consumers and society,” Journal of Business Insight and Innovation, vol. 4, no. 1, pp. 73–87, 2025.

[33] D. Mohiuddin, “Algorithmic hyper-personalization: The double-edged sword of predictive personalization—An empirical investigation,” Journal of Engineering and Computational Intelligence Review, vol. 2, no. 2, pp. 82–94, 2024.

[34] D. Mohiuddin, “Consumer perceptions and trust in AI-generated advertising: An experimental study in the Pakistani context,” Apex Journal of Social Sciences, vol. 3, no. 1, pp. 53–68, 2024.