This paper addresses the problem of obtaining finite dimensional models of distributed-parameter systems from measurement data using system identification. The data are used to construct approximations of the solution and the forcing function in a finite dimensional space, which are expressed in terms of a finite element basis. A discrete time model is estimated based on the resulting finite dimensional coordinate vector. The existence and convergence of such a representation is established for a class of abstract first order systems. The proposed approach is illustrated in practice using simulated noise contaminated data.