| Issue |
ESAIM: M2AN
Volume 60, Number 1, January-February 2026
|
|
|---|---|---|
| Page(s) | 51 - 81 | |
| DOI | https://doi.org/10.1051/m2an/2025095 | |
| Published online | 30 January 2026 | |
Error estimates of asymptotic-preserving neural networks in approximating stochastic linearized Boltzmann equation
The Chinese University of Hong Kong, Hong Kong, P.R. China
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
5
March
2025
Accepted:
25
November
2025
In this paper, we construct an asymptotic-preserving neural network (APNN) (Jin et al., J. Sci. Comput. 94 (2023) 57) for the linearized Boltzmann equation with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss function based on the stochastic-Galerkin system conducted from the micro-macro equations. Rigorous analysis is provided to show the capability of the neural network in approximating solutions near the global Maxwellian. By employing hypocoercivity techniques, we demonstrate two key results: (i) the existence of APNN leading to arbitrarily small loss function, and (ii) the cor convergence of the APNN's approximated solution as the loss tends to zero, with the error exhibiting an exponential decay in time.
Mathematics Subject Classification: 35Q20 / 68T07 / 82C40 / 65F99
Key words: Linearized Boltzmann equation / uncertainty quantification / deep learning / asymptotic-preserving neural networks / hypocoercivity
© The authors. Published by EDP Sciences, SMAI 2026
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