The identification of finite dimensional discrete-time models of deterministic linear and nonlinear infinite dimensional systems from pointwise observations is investigated. The input and output observations are used to construct finite dimensional approximations of the solution and the forcing function which are expanded in terms of a finite element basis. An algorithm to determine a minimal basis to approximate the data is introduced. Subsequently, the resulting coordinate vectors are used to identify a finite dimensional discrete-time model, Theoretical results concerning the existence, stability and convergence of the finite dimensional representation are established. Numerical results involving identification of finite dimensional models for both linear and nonlinear infinite dimensional systems are presented.