Bounded KRnet and its applications to density estimation and
  approximation
Li Zeng Xiaoliang Wan Tao Zhou
In this paper, we develop an invertible mapping, called B-KRnet, on a bounded
domain and apply it to density estimation/approximation for data or the
solutions of PDEs such as the Fokker-Planck equation and the Keller-Segel
equation. Similar to KRnet, the structure of B-KRnet adapts the triangular form
of the Knothe-Rosenblatt rearrangement into a normalizing flow model. The main
difference between B-KRnet and KRnet is that B-KRnet is defined on a hypercube
while KRnet is defined on the whole space, in other words, we introduce a new
mechanism in B-KRnet to maintain the exact invertibility. Using B-KRnet as a
transport map, we obtain an explicit probability density function (PDF) model
that corresponds to the pushforward of a prior (uniform) distribution on the
hypercube. To approximate PDFs defined on a bounded computational domain,
B-KRnet is more effective than KRnet. By coupling KRnet and B-KRnet, we can
also define a deep generative model on a high-dimensional domain where some
dimensions are bounded and other dimensions are unbounded. A typical case is
the solution of the stationary kinetic Fokker-Planck equation, which is a PDF
of position and momentum. Based on B-KRnet, we develop an adaptive learning
approach to approximate partial differential equations whose solutions are PDFs
or can be regarded as a PDF. In addition, we apply B-KRnet to density
estimation when only data are available. A variety of numerical experiments is
presented to demonstrate the effectiveness of B-KRnet.