论文标题
半分析的PINN方法，用于奇异的边界价值问题
Semi-analytic PINN methods for singularly perturbed boundary value problems
论文作者
论文摘要
我们提出了一个新的半分析物理学知情网络（PINN），以解决奇异的边界价值问题。 Pinn是一个科学的机器学习框架，为找到部分微分方程的数值解决方案提供了有希望的观点。 PINN在求解各种微分方程方面表现出令人印象深刻的性能，包括与域复杂几何相关的时间依赖性和多维方程。但是，在考虑僵硬的微分方程时，由于光谱偏置，神经网络通常无法捕获溶液的急剧过渡。为了解决此问题，我们在这里开发了半分析的PINN方法，该方法通过使用从边界层分析获得的所谓校正器函数丰富。我们的新富集的PINN准确地预测了奇异扰动问题的数值解。数值实验包括各种类型的奇异扰动线性和非线性微分方程。
We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical solutions to partial differential equations. The PINNs have shown impressive performance in solving various differential equations including time-dependent and multi-dimensional equations involved in a complex geometry of the domain. However, when considering stiff differential equations, neural networks in general fail to capture the sharp transition of solutions, due to the spectral bias. To resolve this issue, here we develop the semi-analytic PINN methods, enriched by using the so-called corrector functions obtained from the boundary layer analysis. Our new enriched PINNs accurately predict numerical solutions to the singular perturbation problems. Numerical experiments include various types of singularly perturbed linear and nonlinear differential equations.
