We intend to extract an optimal design input of the dynamic computer simulator whose response matches a field observation or a pre-specified target as closely as possible. In this inverse problem, both the field observation and the computer simulator outputs are time series valued. The proposed approach starts by fitting a singular value decomposition based Gaussian process model on an initial design set as a statistical emulator. Subsequently, it selects the follow-up design points by maximizing a novel expected improvement criterion and updates the emulator. Finally, the optimal design input is extracted by the proposed least expected square discrepancy method. The proposed approach is shown to be effective in an illustrative example. Moreover, it achieves better accuracy in extracting the optimal design input compared with existing alternatives in the simulation study and a real application.