Weakly nonlinear oscillators (WNOs) 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 weak nonlinearities. Across these diverse domains, overlooking weak nonlinearities as minor corrections risks substantial errors and system failures, undermining both predictive modeling and real-time control. Identifying weak nonlinearities is essential across fields, from micro-electromechanical systems to large-scale engineering, but remains challenging for data-driven methods due to the profound disparity between nonlinear and dominant harmonic terms. Most data-driven approaches focus on extracting concise and dominant governing equations\cite{SINDy}, often discarding terms deemed “insignificant”. This presents a critical limitation when modeling WNOs, where terms essential to capturing system behavior may be mistakenly excluded, even when clean and complete data are available. The core difficulty lies in the fact that weakly nonlinear terms, by nature, appear subtle in magnitude and are easily filtered out by standard model selection techniques. Here we present EvLOWN, a data-driven approach for inferring the governing equations of WNOs from partial and noisy observations\cite{EvLOWN}. This approach leverages the method of averaging to enable an accurate uncovering of the hidden weakly nonlinear effects, despite their vastly smaller magnitudes. Its effectiveness is demonstrated by the validation of several fundamental oscillatory systems; and its robustness to observational noise is proofed by comparative experiments against nine SOTA algorithms. We apply EvLOWN to two critical engineering systems. First, using publicly available orbital data, we reconstruct the dynamics of Tiangong and International Space Stations, revealing near-identical governing laws despite their distinct missions. Second, we infer the subtle geometry and modal coupling effects of micro-mirror from a single forcing trajectory and predicting full frequency-response curves (FRCs) on different forcing level precisely. These results highlight EvLOWN’s ability to advance data-driven modeling in engineering domains where weak nonlinearities have a significant impact.