A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate the Koopman operator without a clear connection to differential operators, the proposed method constructs finite-dimensional Koopman matrices by projecting the dynamics onto a Chebyshev basis, thereby capturing both differential and nonlinear terms. This establishes a numerical link between the Koopman and differential operators. Numerical experiments on benchmark dynamical systems confirm the accuracy and efficiency of the approach, underscoring its potential for interpretable operator learning. The framework also lays a foundation for future integration with symbolic regression, enabling the construction of explicit mathematical models directly from data.