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The invariant distribution, which is characterized by the stationaryFokker-Planck equation, is an important object in the study of randomlyperturbed dynamical systems. Traditional numerical methods for computing theinvariant distribution based on the Fokker-Planck equation, such as finitedifference or finite element methods, are limited to low-dimensional systemsdue to the curse of dimensionality. In this work, we propose a deep learningbased method to compute the generalized potential, i.e. the negative logarithmof the invariant distribution multiplied by the noise. The idea of the methodis to learn a decomposition of the force field, as specified by theFokker-Planck equation, from the trajectory data. The potential component ofthe decomposition gives the generalized potential. The method can deal withhigh-dimensional systems, possibly with partially known dynamics. Using thegeneralized potential also allows us to deal with systems at low temperatures,where the invariant distribution becomes singular around the metastable states.These advantages make it an efficient method to analyze invariant distributionsfor practical dynamical systems. The effectiveness of the proposed method isdemonstrated by numerical examples.

Computing the Invariant Distribution of Randomly Perturbed Dynamical Systems Using Deep Learning