Background : Machine learning (ML) has emerged as a transformative paradigm in computational and applied mathematics, offering data-driven alternatives to classical numerical solvers and analytical techniques that traditionally rely on domain-specific assumptions.
Objective : This study systematically evaluates leading ML algorithms—neural networks, regression models, and ensemble methods—as tools for solving mathematical problems including partial differential equations, optimization tasks, and function approximation.
Methods : Six algorithms were benchmarked across synthetic and real-world datasets using k-fold cross-validation. Metrics included prediction accuracy, mean squared error (MSE), convergence rate, computational speedup versus classical Finite Element Methods (FEM), and memory usage.
Results : Long Short-Term Memory networks attained the highest predictive accuracy (95.6%) and lowest MSE (0.0071). Physics-Informed Neural Networks (PINNs) achieved an 88.2% error reduction over classical methods with a 12.4× computational speedup. Deep Galerkin Methods reduced error by 91.3% at an 18.7× speedup.
Conclusion : ML algorithms substantially outperform traditional approaches on high-dimensional and nonlinear mathematical problems, offering compelling accuracy–efficiency trade-offs that make them viable candidates for integration into production-grade computational mathematics workflows.