mlsanityData to 7 Oct 2026

Semi-analytic PINN methods for singularly perturbed boundary value problems

arXiv 2208.09145 ↗First posted 19 Aug 2022math.NA cs.LG cs.NAcited by 6

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Abstract (arXiv)

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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 17 of its 40 references that are in the map. First posted on arXiv 19 Aug 2022.

Builds on

17 of 40 references.

References · 17

  1. 2018On the Spectral Bias of Neural Networkscited by 1,321
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  3. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  5. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  6. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  7. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  8. 2019Towards Understanding the Spectral Bias of Deep Learningcited by 173
  9. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  10. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  11. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  12. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
5 more
  1. 2020Physics-informed neural network for ultrasound nondestructive quantification of surface breaking crackscited by 67
  2. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  3. 2020On generalized residue network for deep learning of unknown dynamical systemscited by 16
  4. 2020Weak Form Theory-guided Neural Network (TgNN-wf) for Deep Learning of Subsurface Single and Two-phase Flowcited by 12
  5. 2021Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networkscited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 2; 2026: 3.

6 papers.

  1. 2024VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behaviorcited by 13
  2. 2025Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradientscited by 7
  3. 2025Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEscited by 1
  4. 2026Soft Partition-based KAPI-ELM for Multi-Scale PDEscited by 0
  5. 2026A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networkscited by 0
  6. 2026Selective boundary condition reduction via learned error gatingcited by 0