A Bi-fidelity based asymptotic-preserving neural network for the semiconductor Boltzmann equation and its inverse problem

2025/11/17 by Liu, Liu, Zhu, Xueyu, Zhu, Zhenyi
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2511.13321

Abstract

This paper introduces a Bi-fidelity Asymptotic-Preserving Neural Network (BI-APNNs) framework, designed to efficiently solve forward and inverse problems for the semiconductor Boltzmann equation. Our approach builds upon the Asymptotic-Preserving Neural Network (APNNs) methodology \citeAPNN-transport, which employs a micro-macro decomposition to handle the model's multiscale nature. We specifically address a key bottleneck in the original APNNs: the slow convergence of the macroscopic density ρ in the near fluid-dynamic regime, i.e., for small Knudsen numbers ε. The core innovation of BI-APNNs is a novel bi-fidelity decomposition of the macroscopic quantity ρ, which accurately approximates the true density at small ε, and can be efficiently pre-trained. A separate and more compact neural network is then tasked with learning only the minor correction term, ρ_corr. This strategy not only significantly \it accelerates the training convergence but also improves the accuracy of the forward problem solution, particularly in the challenging fluid-dynamic limit. Meanwhile, we demonstrate through extensive numerical experiments that our new BI-APNNs yields substantially more accurate and robust results for inverse problems compared to the standard APNNs. Validated on both the semiconductor Boltzmann and the Boltzmann-Poisson systems, our work shows that the bi-fidelity formulation is a powerful enhancement for tackling multiscale kinetic equations, especially when dealing with inverse problems constrained by partial observation data.

Citations

• PhysicsSolver: Transformer-Enhanced Physics-Informed Neural Networks for Forward and Forecasting Problems in Partial Differential Equations
• Asymptotic-preserving neural networks for the semiconductor Boltzmann equation and its application on inverse problems
• Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations
• Deep adaptive basis Galerkin method for high-dimensional evolution equations with oscillatory solutions
• Asymptotic-Preserving Neural Networks for Multiscale Time-Dependent Linear Transport Equations
• Reconstruction of the collision kernel in the nonlinear Boltzmann equation
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Survey of multifidelity methods in uncertainty propagation, inference,\n and optimization
• Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization
• Adam: A Method for Stochastic Optimization
• Kinetic description of optimal control problems and applications to\n opinion consensus
• Multilayer feedforward networks are universal approximators

Related

• Asymptotic-preserving neural networks for the semiconductor Boltzmann equation and its application on inverse problems
• Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations
• Error estimates of physics-informed neural networks for approximating Boltzmann equation
• An Investigation of Physics Informed Neural Networks to solve the Poisson-Boltzmann Equation in Molecular Electrostatics
• Error estimates of asymptotic-preserving neural networks in approximating stochastic linearized Boltzmann equation