  We study a class of nonlinear nonparametric inverse problems. Specifically,
we propose a nonparametric estimator of the dynamics of a monotonically
increasing trajectory defined on a finite time interval. Under suitable
regularity conditions, we prove consistency of the proposed estimator and show
that in terms of $L^2$-loss, the optimal rate of convergence for the proposed
estimator is the same as that for the estimation of the derivative of a
trajectory. This is a new contribution to the area of nonlinear nonparametric
inverse problems. We conduct a simulation study to examine the finite sample
behavior of the proposed estimator and apply it to the Berkeley growth data.
