ASPINN: An asymptotic strategy for solving singularly perturbed differential equations
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Abstract (arXiv)
Solving Singularly Perturbed Differential Equations (SPDEs) presents challenges due to the rapid change of their solutions at the boundary layer. In this manuscript, We propose Asymptotic Physics-Informed Neural Networks (ASPINN), a generalization of Physics-Informed Neural Networks (PINN) and General-Kindred Physics-Informed Neural Networks (GKPINN) approaches. This is a decomposition method based on the idea of asymptotic analysis. Compared to PINN, the ASPINN method has a strong fitting ability for solving SPDEs due to the placement of exponential layers at the boundary layer. Unlike GKPINN, ASPINN lessens the number of fully connected layers, thereby reducing the training cost more effectively. Moreover, ASPINN theoretically approximates the solution at the boundary layer more accurately, which accuracy is also improved compared to GKPINN. We demonstrate the effect of ASPINN by solving diverse classes of SPDEs, which clearly shows that the ASPINN method is promising in boundary layer problems. Furthermore, we introduce Chebyshev Kolmogorov-Arnold Networks (Chebyshev-KAN) instead of MLP, achieving better performance in various experiments.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 20 of its 38 references that are in the map. First posted on arXiv 20 Sep 2024.
Builds on
20 of 38 references.
References · 20
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
- 2024KAN: Kolmogorov-Arnold Networkscited by 634
- 2019DeepXDE: A deep learning library for solving differential equationscited by 617
- 2016Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networkscited by 541
- 2021Neural Fields in Visual Computing and Beyondcited by 539
- 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
- 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
- 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
- 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
- 2024Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximationcited by 115
- 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
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- 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 103
- 2023Residual-based attention in physics-informed neural networkscited by 72
- 2021Learning in Sinusoidal Spaces with Physics-Informed Neural Networkscited by 61
- 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
- 2020A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecturecited by 50
- 2021Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusioncited by 38
- 2024RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networkscited by 5
- 2024General-Kindred Physics-Informed Neural Network to the Solutions of Singularly Perturbed Differential Equationscited by 2
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Citing papers per year: 2025: 1; 2026: 1.
2 papers.