Year: 2025
Type: article
Abstract: . 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, B-KRnet consists of a series of coupling layers with progressively fewer active transformation dimensions, inspired by the triangular structure of the Knothe–Rosenblatt (KR) rearrangement. 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, a new mechanism is introduced 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 base (uniform) distribution on the hypercube. It can be directly applied to density estimation when only data are available. By coupling KRnet and B-KRnet, we 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 treated as PDFs. A variety of numerical experiments is presented to demonstrate the effectiveness of B-KRnet.
DOI: 10.1137/23m1579091
Authors: Li Zeng, Xiaoliang Wan, Tao Zhou
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416615263