Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. In many cases, time series output can be usefully reduced via principal components to simplify analysis. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. However, inputs that are functions of time often do not have one or a few “characteristic shapes” that are more common with output functions, and so, principal component representation has less potential for reducing the dimension of input functions. In this article, Gaussian process surrogates are described for models with inputs and outputs that are both functions of time. The focus is on construction of an appropriate covariance structure for such surrogates, some experimental design issues, and an application to a model of marrow cell dynamics.