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The invariant distribution is an important object in the study of randomly perturbed dynamical systems. The existing methods, including traditional finite difference or finite element methods as well as recently developed machine learning-based methods, all require the knowledge of the dynamical equations or adequate equilibrium data for estimating the invariant distribution. In this work, we propose a data-driven method for inferring the dynamics and simultaneously learning the invariant distribution from noisy trajectory data of the dynamical system. The data is not necessarily at equilibrium and may be collected from the transient period of the dynamics. The proposed method combines the idea of maximum likelihood estimation and a decomposition of the force field as suggested by the Fokker-Planck equation. The drift term (or force field), which is in the form of a decomposition with the constraint specified by the …