Deforming them is tricky and the deformations must be integrated into the actual model. That is, without something like a non-linear deformation space, as this paper talks about.

Lawrence J. Christiano, Solving the Stochastic Growth Model by Linear-Quadratic Approximation and by Value-Function Iteration, Journal of Business & Economic Statistics, Vol. 8, No. 1 (Jan., 1990), pp ...

Nicole Malfait, James O. Ramsay, The Historical Functional Linear Model, The Canadian Journal of Statistics / La Revue Canadienne de Statistique, Vol. 31, No. 2 (Jun ...