mlsanityData to 14 Oct 2026

High precision PINNs in unbounded domains: application to singularity formulation in PDEs

arXiv 2506.19243 ↗First posted 24 Jun 2025cs.LG cs.NA math.NAcited by 7

Not on the map. Why?

Abstract (arXiv)

We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is 4 digits smaller than that obtained in citewang2023asymptotic with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 33 of its 64 references that are in the map. First posted on arXiv 24 Jun 2025.

Builds on

33 of 64 references.

References · 33

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  3. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  5. 2018On the Spectral Bias of Neural Networkscited by 1,341
  6. 2024KAN: Kolmogorov-Arnold Networkscited by 649
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  8. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  9. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  10. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 368
  11. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 303
  12. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
21 more
  1. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  2. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  3. 2018Training behavior of deep neural network in frequency domaincited by 174
  4. 2023The Quantization Model of Neural Scalingcited by 138
  5. 2024KAN 2.0: Kolmogorov-Arnold Networks Meet Sciencecited by 118
  6. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  7. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 104
  8. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  9. 2023Residual-based attention in physics-informed neural networkscited by 72
  10. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
  11. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 56
  12. 2023Multi-stage Neural Networks: Function Approximator of Machine Precisioncited by 48
  13. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  14. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 38
  15. 2024On the expressiveness and spectral bias of KANscited by 37
  16. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  17. 2022On the Activation Function Dependence of the Spectral Bias of Neural Networkscited by 28
  18. 2024Gauss-Newton Natural Gradient Descent for Physics-Informed Computational Fluid Dynamicscited by 28
  19. 2022Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networkscited by 19
  20. 2022FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuationcited by 13
  21. 2024Spectrum-Informed Multistage Neural Networks: Multiscale Function Approximators of Machine Precisioncited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 3; 2026: 4.

7 papers.

  1. 2025ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithmscited by 7
  2. 2026Fourier Feature Pyramids for Physics-Informed Neural Networkscited by 1
  3. 2025Automated discovery of finite volume schemes using Graph Neural Networkscited by 0
  4. 2025Evaluation of Sampling Strategies and Physics-Informed Kolmogorov--Arnold Networks in Unbounded Domainscited by 0
  5. 2026Solving nonlinear subsonic compressible flow in infinite domain via multi-stage neural networkscited by 0
  6. 2026Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approachescited by 0
  7. 2026SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learningcited by 0