  The invariant distribution, which is characterized by the stationary
Fokker-Planck equation, is an important object in the study of randomly
perturbed dynamical systems. Traditional numerical methods for computing the
invariant distribution based on the Fokker-Planck equation, such as finite
difference or finite element methods, are limited to low-dimensional systems
due to the curse of dimensionality. In this work, we propose a deep learning
based method to compute the generalized potential, i.e. the negative logarithm
of the invariant distribution multiplied by the noise. The idea of the method
is to learn a decomposition of the force field, as specified by the
Fokker-Planck equation, from the trajectory data. The potential component of
the decomposition gives the generalized potential. The method can deal with
high-dimensional systems, possibly with partially known dynamics. Using the
generalized 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 distributions
for practical dynamical systems. The effectiveness of the proposed method is
demonstrated by numerical examples.
