The augmented, iterated Kalman smoother is applied to system identification for inverse problems in evolutionary differential equations. In the augmented smoother, the unknown, time-dependent coefficients are included in the state vector, and have a stochastic component. At each step in the iteration, the estimate of the time evolution of the coefficients is linear. We update the slowly varying mean temperature and conductivity by averaging the estimates of the Kalman smoother. Applications include the estimation of anomalous diffusion coefficients in turbulent fluids and the plasma rotation velocity in plasma tomography.