Problem
Research questions and friction points this paper is trying to address.
Develops B-KRnet for bounded domain density estimation
Approximates PDE solutions like Fokker-Planck equation
Combines KRnet and B-KRnet for mixed-dimension modeling
Bounded KRnet and its applications to density estimation and approximation
🤖 AI Summary
This work addresses the challenge of probability density function (PDF) estimation on bounded domains and PDF modeling of partial differential equation (PDE) solutions. We propose B-KRnet, the first strictly invertible, pseudo-triangular normalizing flow defined on the unit hypercube. Methodologically, we design a boundary-adapted invertible mapping architecture enabling exact pushforward density modeling, and introduce a coupling mechanism between KRnet and B-KRnet to support hybrid bounded/unbounded high-dimensional generation. Our key contributions are: (i) the first strictly invertible pseudo-triangular flow on bounded domains—eliminating truncation artifacts and boundary leakage; and (ii) an adaptive PDE solver framework that directly learns the joint distribution (e.g., position-momentum) of solutions to Fokker–Planck and Keller–Segel equations. Numerical experiments demonstrate substantial improvements in bounded-domain density estimation accuracy and establish state-of-the-art performance in high-dimensional dynamical PDF approximation.
📝 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, the structure of B-KRnet adapts the pseudo-triangular structure 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, 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 prior (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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
B-KRnet: invertible mapping on bounded domain
Coupling layers with fewer active dimensions
Adaptive learning for PDE solutions as PDFs
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