Dynamical systems arise in a wide variety of mathematical models from science\nand engineering. A common challenge is to quantify uncertainties on model\ninputs (parameters) that correspond to a quantitative characterization of\nuncertainties on observable Quantities of Interest (QoI). To this end, we\nconsider a stochastic inverse problem (SIP) with a solution described by a\npullback probability measure. We call this an observation-consistent solution,\nas its subsequent push-forward through the QoI map matches the observed\nprobability distribution on model outputs. A distinction is made between QoI\nuseful for solving the SIP and arbitrary model output data. In dynamical\nsystems, model output data are often given as a series of state variable\nresponses recorded over a particular time window. Consequently, the dimension\nof output data can easily exceed $\\mathcal{O}(1E4)$ or more due to the\nfrequency of observations, and the correct choice or construction of a QoI from\nthis data is not self-evident. We present a new framework, Learning Uncertain\nQuantities (LUQ), that facilitates the tractable solution of SIPs for dynamical\nsystems. Given ensembles of predicted (simulated) time series and (noisy)\nobserved data, LUQ provides routines for filtering data, unsupervised learning\nof the underlying dynamics, classifying observations, and feature extraction to\nlearn the QoI map. Subsequently, time series data are transformed into samples\nof the underlying predicted and observed distributions associated with the QoI\nso that solutions to the SIP are computable. Following the introduction and\ndemonstration of LUQ, numerical results from several SIPs are presented for a\nvariety of dynamical systems arising in the life and physical sciences. For\nscientific reproducibility, we provide links to our Python implementation of\nLUQ and to all data and scripts required to reproduce the results in this\nmanuscript.\n