2023/05/15 by Zeng Li, Xiaoliang Wan, Zeng, Li +3 · 1 citation
Computer Science · Physics and Astronomy · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning and Algorithms #Model Reduction and Neural Networks #Neural Networks and Applications
paper · pdf · doi:10.48550/arxiv.2305.09063
openalex publication_date 2023/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.