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.
翻译:我们提出一个新的半分析物理神经信息网络(PINN),以解决异常扰动的边界值问题。 PINN是一个科学机器学习框架,为寻找局部差异方程式的数字解决方案提供了一个有希望的视角。 PINN在解决各种差异方程式(包括复杂的域内几何中涉及的时间依赖和多维方程式)方面表现出了令人印象深刻的成绩。然而,在考虑硬度差异方程式时,神经网络一般无法捕捉解决方案的急剧转变,因为光谱偏差。为了解决这个问题,我们在这里开发了半分析型PINN方法,通过利用从边界层分析中获得的所谓正确函数加以补充。我们新的丰富PINNs准确地预测了单振动问题的数字解决方案。数字实验包括各种单振动线性和非线性差异方程式。