SOBOLEV SPACES AND THE VARIATIONAL THEORY OF OPTIMAL QUADRATURE FORMULAS FOR EVALUATING HIGHLY OSCILLATORY INTEGRALS: NEW APPROACHES AND PRACTICAL APPLICATIONS

This paper explores the variational principles of constructing optimal quadrature formulas with derivatives in Sobolev-Hilbert spaces for evaluating highly oscillatory integrals. Criteria of Hermite interpolation and Sard optimality, incorporating both function values and higher-order derivatives, are systematically analyzed. As an innovation, this study introduces Machine Learning-driven spline approximation and adaptive neural network integration algorithms (PINNs) to optimize node localization. The practical applications of the obtained formulas in radiophysics, quantum mechanics, and seismology are illustrated, backed by rigorous error estimates of order .