PARAMETRIZATION METHOD AND PHYSICS-INFORMED NEURAL NETWORKS FOR THE DUFFING EQUATION

Abstract

Introduction. Physics-informed neural networks (PINNs) are increasingly proposed as universal solvers of differential equations, but their reliability for nonlinear oscillators with several periodic solutions is insufficiently studied. Aims and objectives. To compare D.S. Dzhumabaev’s parametrization method with PINNs for periodic solutions of the Duffing equation in terms of accuracy, cost, robustness and control over coexisting solutions. Results. The parametrization method reached errors of 10⁻⁸-10⁻⁷ in hundredths of a second; eight subintervals raised the share of convergent initial approximations under strong softening nonlinearity from 0-9% to 61-100%; with arclength continuation and Floquet multipliers, the method reproduced the full hysteresis of the frequency response. PINNs reached errors of 10⁻⁶-10⁻² in 20-40 s and in some regimes converged, with an equally small loss, to a different periodic solution. Conclusion. PINN solutions require verification, which the parametrization method provides.