Physically‑Informed Segmented Optimization for Full‑Range Frequency‑Amplitude Modelling of Nonlinear Oscillators

Accurate frequency‑amplitude relations are essential for analysing free vibration, forced resonance, and synchronization in nonlinear oscillators. However, conventional perturbation and linearized approximations often incur large errors under strong nonlinearity and cannot capture large‑amplitude stiffness‑hardening behaviour. This work proposes **Physics‑Informed Segmented Optimization (PISO)**, a physics‑constrained hybrid analytical‑optimization framework for full‑range frequency‑amplitude modelling of conservative nonlinear oscillators. Using an analytical square‑root ansatz with even‑power amplitude terms, the method divides the amplitude domain into ultra‑small, conventional, and large‑amplitude regimes governed by competing linear and nonlinear restoring forces. Regime‑specific coefficients are identified by minimizing the relative root‑mean‑square error against exact reference natural frequencies. Benchmark tests on cubic and ninth‑order Duffing oscillators show maximum relative prediction errors below 0.477% and 1% across the full amplitude range, with effective suppression of phase distortion. Coupled with the harmonic‑balance method, PISO also predicts bistable hysteresis and jump bifurcations in damped forced‑vibration responses, with amplitude deviations below 0.5% relative to direct numerical integration. In NEMS synchronization, the method captures physical saturation in synchronization bandwidth and noise suppression, avoiding the unphysical unbounded growth produced by standard linear approximations. Overall, PISO provides an explicit, computationally efficient, and physically consistent analytical tool for nonlinear‑oscillator analysis, with practical value for NEMS resonator design.