Solving multiscale steady radiative transfer equation using neural networks with uniform stability
Solving multiscale steady radiative transfer equation using neural networks with uniform stability
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DOI：
10.1007/s40687-022-00345-z
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发表时间：
2021-10
影响因子：
1.2
通讯作者：
Yulong Lu;Li Wang;Wuzhe Xu
中科院分区：
文献类型：
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作者：
Yulong Lu;Li Wang;Wuzhe Xu
利用物理信息神经网络（PINNs）求解含扩散标度的稳态辐射传输方程。PINN的思想是最小化最小二乘损失函数，该函数由控制方程的残差、边界条件的失配和其他物理约束（如守恒）组成。它的优点是灵活和易于执行，并带来了潜在的高维问题。然而，由于小尺度的存在，香草PINN可以是非常不稳定的解决多尺度稳态传输方程。在本文中，我们提出了一个新的公式的损失的基础上的宏微观分解。我们证明了新的损失函数是一致稳定的关于小Knudsen数的意义下，神经网络的解决方案的误差是一致控制的损失。当边界条件为各向异性时，在扩散极限处会出现边界层，从而给神经网络的训练带来额外的困难。为了解决这个问题，我们包括一个边界层校正器，它可以保留解的急剧过渡部分，并使其余部分易于近似。新方法的有效性证明在广泛的数值例子。
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-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.