• SUMIT KUMAR SRIVASTAVA

The growing integration of machine learning with mathematical computation has introduced new approaches for approximating and solving complex problems. This study investigates its applicability to nonlinear algebraic equations, ordinary differential equations, nonlinear function approximation, and mathematical optimization. Conventional procedures are compared with supervised and residual-based neural-network models in terms of accuracy, convergence, generalization, and computational relevance. For nonlinear equations, Newton-Raphson iteration produced a highly accurate solution for an individual cubic, whereas a multilayer perceptron learned a reusable coefficient-to-root mapping for 2,000 depressed cubic equations. The model achieved a test MAE of 0.008862, RMSE of 0.025214, and R² of 0.999538. For the initial-value problem y′ = y, y(0) = 1, a residual-based neural trial solution closely reproduced the exact solution y = eˣ, with an RMSE of 7.19 × 10⁻⁹. By comparison, Euler's method with h = 0.1 produced an absolute error of approximately 0.124539 at x = 1. A supervised neural network approximating sin(x) over [0, 2π] achieved a test MAE of 0.002098, RMSE of 0.002529, and R² of 0.999988. An analytical optimization benchmark was additionally used to formulate a verifiable framework for surrogate-based optimization. The results show that machine-learning models can provide accurate and reusable approximations when mathematical tasks involve repeated evaluations, nonlinear mappings, or governing differential constraints. Their reliability, however, depends on the data, architecture, optimization procedure, problem domain, and validation criteria. Conventional methods remain preferable for many simple and well-structured problems because of their precision, transparency, and theoretical guarantees. Machine learning is therefore most appropriately viewed as a complementary computational methodology rather than a universal replacement for established analytical and numerical techniques.

Machine Learning; Mathematical Problem Solving; Artificial Neural Networks; Nonlinear Algebraic Equations; Ordinary Differential Equations; Function Approximation; Numerical Analysis; Surrogate Optimization.

"Application Of Machine Learning In Mathematical Problem Solving: Experimental Results, Analysis And Interpretation", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.13, Issue 9, page no.b313-b337, September-2026, Available :http://www.jetir.org/papers/JETIR2609137.pdf

"Application Of Machine Learning In Mathematical Problem Solving: Experimental Results, Analysis And Interpretation", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.13, Issue 9, page no. ppb313-b337, September-2026, Available at : http://www.jetir.org/papers/JETIR2609137.pdf