Strongly nonlinear systems, which commonly arise in turbulent flows and climate dynamics, are characterized by persistent and intermittent energy transfer between various spatial and temporal scales. These systems are difficult to model and analyze due to either high-dimensionality or uncertainty and there has been much recent interest in obtaining reduced models, for example in the form of stochastic closures, that can replicate their non-Gaussian statistics in many dimensions. On the other hand, data-driven methods, powered by machine learning and operator theoretic concepts, have shown great utility in modeling nonlinear dynamical systems with various degrees of complexity. Here we propose a data-driven framework to model stationary chaotic dynamical systems through non-linear transformations and a set of decoupled stochastic differential equations (SDEs). Specifically, we first use optimal transport to find a transformation from the distribution of time-series data to a multiplicative reference probability measure such as the standard normal distribution. Then we find the set of decoupled SDEs that admit the reference measure as the invariant measure, and also closely match the spectrum of the transformed data. As such, this framework represents the chaotic time series as the evolution of a stochastic system observed through the lens of a nonlinear map. We demonstrate the application of this framework in Lorenz-96 system, a 10-dimensional model of high-Reynolds cavity flow, and reanalysis climate data. These examples show that SDE models generated by this framework can reproduce the non-Gaussian statistics of systems with moderate dimensions (e.g. 10 and more), and approximate super-Gaussian tails that are not readily computable from the training data. Some advantageous features of our framework are convexity of the main optimization problem, flexibility in choice of reference measure and the SDE model, and interpretability in terms of interaction between variables and the underlying dynamical process.