[Paper Review] Solving multiscale steady radiative transfer equation using neural networks with uniform stability

Yulong Lu, Li Wang+1 more|arXiv (Cornell University)|Oct 13, 2021

Mathematical Biology Tumor Growth4 citations

TL;DR

This paper proposes a physics-informed neural network (PINN) framework with a macro-micro decomposition-based loss function to solve the multiscale steady radiative transfer equation (RTE) with uniform stability in the diffusive regime (small Knudsen number ε). By reformulating the loss using the macro-micro decomposition and incorporating a boundary layer corrector for anisotropic inflow conditions, the method ensures that the L²-error of the neural network solution is uniformly bounded with respect to ε, resolving the instability issues of vanilla PINNs in the ε → 0 limit.

ABSTRACT

This paper concerns solving the steady radiative transfer equation with diffusive scaling, using the physics informed neural networks (PINNs). The idea of PINNs is to minimize a least-square loss function, that consists of the residual from the governing equation, the mismatch from the boundary conditions, and other physical constraints such as conservation. It is advantageous of being flexible and easy to execute, and brings the potential for high dimensional problems. Nevertheless, due the presence of small scales, the vanilla PINNs can be extremely unstable for solving multiscale steady transfer equations. In this paper, we propose a new formulation of the loss based on the macro-micro decomposition. We prove that, the new loss function is uniformly stable with respect to the small Knudsen number in the sense that the $L^2$-error of the neural network solution is uniformly controlled by the loss. When the boundary condition is an-isotropic, a boundary layer emerges in the diffusion limit and therefore brings an additional difficulty in training the neural network. To resolve this issue, we include a boundary layer corrector that carries over the sharp transition part of the solution and leaves the rest easy to be approximated. The effectiveness of the new methodology is demonstrated in extensive numerical examples.

Motivation & Objective

• To address the instability of vanilla physics-informed neural networks (PINNs) when solving the steady radiative transfer equation (RTE) in the diffusive regime (small ε).
• To develop a loss function that maintains uniform stability with respect to the Knudsen number ε, ensuring the L²-error of the solution remains bounded as ε → 0.
• To handle the challenge of boundary layers arising from anisotropic inflow conditions in the diffusive limit, which hinder neural network convergence.
• To construct a boundary layer corrector based on the half-space problem to isolate sharp transitions and improve approximation of smooth solution components.
• To demonstrate the effectiveness of the proposed method through extensive numerical experiments across linear and nonlinear RTE problems.

Proposed method

• Formulate a new loss function based on the macro-micro decomposition of the RTE solution, separating the solution into a macroscopic (diffusive) part and a microscopic (deviation) part.
• Define the loss as the sum of residuals from the governing equation, boundary conditions, and physical constraints, all expressed in terms of the macro-micro variables.
• Introduce a boundary layer corrector derived from the solution to the associated half-space problem to capture sharp gradients near boundaries in the anisotropic case.
• Use the corrector to decompose the solution into a part with sharp transitions (handled by the corrector) and a smooth residual part (approximated by the neural network).
• Train the neural network to approximate the smooth residual part while enforcing the corrector structure, ensuring stability and accuracy in the ε → 0 limit.
• Prove a uniform stability estimate: the L²-error of the PINN solution is bounded by the new loss function uniformly in ε, independent of the Knudsen number.

Experimental results

Research questions

• RQ1 Can a PINN loss function be designed such that its stability is uniform with respect to the Knudsen number ε in the diffusive regime?
• RQ2 How can boundary layers induced by anisotropic inflow conditions be effectively handled in PINN-based solutions of the RTE?
• RQ3 Does the macro-micro decomposition lead to a more stable and accurate neural network solution for multiscale radiative transfer problems?
• RQ4 Can a boundary layer corrector improve convergence and reduce error in PINN training for problems with sharp gradients near boundaries?
• RQ5 Is the proposed method effective for both linear and nonlinear variants of the steady radiative transfer equation?

Key findings

• The proposed loss function ensures uniform stability: the L²-error of the neural network solution is bounded by the loss uniformly in ε, preventing blow-up as ε → 0.
• For anisotropic boundary conditions, the inclusion of a boundary layer corrector significantly improves training convergence and solution accuracy by isolating sharp gradients.
• Numerical results show that the method achieves high accuracy for both ε = 1 (kinetic regime) and ε = 10⁻³ (diffusive regime), with relative L² errors remaining low across scales.
• In Example 8 (linear RTE), the method achieves accurate approximation of the solution and its diffusion limit, with the loss and error decreasing steadily during training.
• In Example 9 (nonlinear RTE), the method successfully captures the transition from kinetic to diffusive behavior, with the predicted T(x) and I(x,v) closely matching the reference solutions at both ε = 1 and ε = 10⁻³.
• The use of the boundary layer corrector leads to a more stable and efficient training process, especially in the ε → 0 limit, compared to vanilla PINNs.