In fluid flow problems, complex dynamics, manifested in terms of large and small scale spatiotemporal features, is commonly encountered. Resolution of such dynamics results in high-dimensional discretized numerical models with associated high computational cost. This renders the full-scale models intractable in applications where repeated realizations are required, such as flow control. Reduced-order model (ROM) offers a way to mitigate this issue by offering a low-dimensional system which is computationally efficient and accurate. An automated data-driven inference of ROMs circumvents the requirement of an a priori knowledge of the governing equations by relying on the time-series data obtained from simulation or experiments for the modeling. In this paper, a nonintrusive approach based on artificial neural network (NN-ROM) is considered for the reduced-order modeling. NN-ROM serves as a map approximating the reduced coefficients of a high-fidelity solution in low-dimensional space. A novel multistep, residual-based neural network framework is proposed. The proposed approach is used to recover the dynamical states in numerical and experimental fluid flow problems. The framework provides sufficiently accurate initial estimate. Deviations in the long-term prediction are mitigated by augmenting the framework with data assimilation (DA).