Modeling of the parabolic distributed parameter system (DPS) with the Karhunen-Loeve (KL) method under sparse sensing will become very difficult because the information from the measurements is incomplete. A novel information completion and learning strategy is proposed for spatiotemporal modeling under sparse sensing. During the offline initialization phase, the initial full spatial basis functions (SBFs) are constructed first in the full sensing environment under the framework of time-space separation. Subsequently, during the normal operation phase of fewer sensors, the sparse SBFs are obtained and further used to complete the lost spatial information with the help of the initial full SBFs, which are then recursively calibrated by the incremental KL. By iteratively repeating these two steps, the sparse spatiotemporal output can be completed in a streaming data environment. Finally, the proper spatiotemporal model can be constructed through time-space synthesis. The experimental results of a nonlinear transport-reaction process on a catalytic rod demonstrate the effectiveness of the proposed method.