Semi-analytic PINN methods for singularly perturbed boundary value problems
This work addresses a specific problem in scientific machine learning for researchers and practitioners dealing with stiff differential equations, representing an incremental improvement by enhancing existing PINN methods with analytical insights.
The authors tackled the challenge of solving singularly perturbed boundary value problems using physics-informed neural networks (PINNs), which often fail to capture sharp transitions due to spectral bias, by developing a semi-analytic PINN method enriched with corrector functions from boundary layer analysis, resulting in accurate predictions for various linear and nonlinear differential equations.
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.