This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional PN closure. However, the ML moment closure model in \cite{huang2021gradient} is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model.
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This content will become publicly available on July 1, 2025
The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients
This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping
coefficient. Our approach involves approximating the hyperbolic equation’s solution by its truncated Fourier
expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation
process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic
equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman
contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The
method not only delivers precise solutions but also showcases remarkable computational efficiency.
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- Award ID(s):
- 2208159
- NSF-PAR ID:
- 10510713
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Computers & Mathematics with Applications
- Volume:
- 166
- Issue:
- C
- ISSN:
- 0898-1221
- Page Range / eLocation ID:
- 77 to 90
- Subject(s) / Keyword(s):
- Time reduction Carleman contraction mapping Initial condition Damping coefficient
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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