  In this work, we propose adaptive deep learning approaches based on
normalizing flows for solving fractional Fokker-Planck equations (FPEs). The
solution of a FPE is a probability density function (PDF). Traditional
mesh-based methods are ineffective because of the unbounded computation domain,
a large number of dimensions and the nonlocal fractional operator. To this end,
we represent the solution with an explicit PDF model induced by a flow-based
deep generative model, simplified KRnet, which constructs a transport map from
a simple distribution to the target distribution. We consider two methods to
approximate the fractional Laplacian. One method is the Monte Carlo
approximation. The other method is to construct an auxiliary model with
Gaussian radial basis functions (GRBFs) to approximate the solution such that
we may take advantage of the fact that the fractional Laplacian of a Gaussian
is known analytically. Based on these two different ways for the approximation
of the fractional Laplacian, we propose two models, MCNF and GRBFNF, to
approximate stationary FPEs and MCTNF to approximate time-dependent FPEs. To
further improve the accuracy, we refine the training set and the approximate
solution alternately. A variety of numerical examples is presented to
demonstrate the effectiveness of our adaptive deep density approaches.
