Encoding Cumulation to Learn Perturbative Nonlinear Oscillatory Dynamics.

Oscillatory dynamics play a central role in the description of a broad spectrum of physical systems. While often well-approximated by linear models, the essential long-term evolution and stability of these systems are frequently determined by subtle nonlinearities. The characterization of such weakly nonlinear systems from observational data is a central challenge in system identification, a task made difficult by the immense disparity in magnitude between faint nonlinear signatures and the dominant linear response. Herein, we introduce Evolutionary Learning Oscillator with Weak Nonlinearity (EvLOWN), a data-driven methodology for inferring the governing equations of weakly nonlinear oscillators from sparse and potentially noisy time-series observations. We first demonstrate EvLOWN's superior accuracy and robustness on benchmark systems, then apply it to uncover the subtle on-site and coupling potentials in fundamental models, including the Fermi-Pasta-Ulam and Klein-Gordon chains, using only local measurements. Translating this framework to critical engineering applications, we reconstruct the orbital dynamics of the Tiangong and International Space Stations from public data, revealing nearly identical governing laws despite their distinct architectures. Furthermore, from wind-tunnel experiments on a scaled suspension bridge, EvLOWN extracts the precise equations of motion capturing complex vortex-induced vibrations. These results establish EvLOWN as a powerful tool for the data-driven discovery of governing laws in complex systems where weak nonlinearities play a crucial yet subtle role.