In this paper we develop a neural network for the numerical simulation of time-dependent linear transport equations with diffusive scaling and uncertainties. The goal of the network is to resolve the computational challenges of curse-of-dimensionality and multiple scales of the problem. We first show that a standard Physics-Informed Neural Network (PINNs) fails to capture the multiscale nature of the problem, hence justifies the need to use Asymptotic-Preserving Neural Networks (APNNs). We show that not all classical AP formulations are fit for the neural network approach. We construct a micro-macro decomposition based neutral network, and also build in a mass conservation mechanism into the loss function, in order to capture the dynamic and multiscale nature of the solutions. Numerical examples are used to demonstrate the effectiveness of this APNNs.
翻译:在本文中,我们开发了一个神经网络,用于对具有不同尺寸和不确定性的时间依赖线性传输方程式进行数字模拟。网络的目标是解决多层面诅咒和问题多重尺度的计算挑战。我们首先表明,标准的物理成形神经网络(PINNs)未能捕捉到问题的多尺度性质,因此证明有必要使用Asymptistic-Pavisive NealNetworks(APNNSs) 。我们表明,并非所有传统的AP方程式都适合神经网络方法。我们建造了一个基于微型-宏观的中性分解网,并将一个大规模保护机制纳入损失功能,以便捕捉解决办法的动态和多尺度性质。我们用一些数字例子来证明这种APNNs的有效性。