Before revision:  Data in biology are often sampled sparsely in time , therefore a method is developed to compute a temporally fine-grained Koopman operator from the temporally coarse-grained Koopman operator . In the case of noisy data, a closed form expression for the error in the coarse-grained Koopman operator is derived. A novel algorithm for optimal sensor placement is developed for the case of a fixed initial condition. The method is finally demonstrated on a simulation example  .
After revision:  Data in biology are often sampled sparsely in time , therefore a method is developed to compute a temporally fine-grained Koopman operator from the temporally coarse-grained Koopman operator , the latter of which is identified from the sparse data. The optimal placement of sensors then corresponds to maximizing the observability of the fine-grained system. We demonstrate the algorithm on a simulation example .