<p>Classical high-order weighted essentially non-oscillatory (WENO) schemes are designed to achieve optimal convergence order for smooth solutions and to maintain non-oscillatory behaviors for discontinuities. However, their spectral properties are not optimal, which limits the ability to capture high-frequency waves and small-scale features. To improve the dispersion and dissipation of the WENO schemes, a data-driven optimized WENO method is proposed. Firstly, we conduct an analysis on the spectral error of nonlinear schemes and derive an explicit upper bound, which provides a theoretical foundation on how the loss function associated with the spectral property is defined in the data-driven optimized scheme. Meanwhile, the total variation diminishing (TVD) constraint and anti-dissipation penalization are incorporated into the loss function to suppress oscillations near discontinuities and preserve stability in simulating high-frequency waves. The results of the benchmark cases demonstrate that the new schemes maintain the shock-capturing capability while providing higher resolution for fine-scale flow features. The approximate dispersion relation (ADR) indicates that the new schemes can match the exact spectrum more accurately over a broader range of wavenumbers.</p>