mlsanityData to 7 Oct 2026

Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks

arXiv 2201.06780 ↗First posted 18 Jan 2022math.AP cs.LG physics.flu-dyncited by 19

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Abstract (arXiv)

Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the C'ordoba-C'ordoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 1 of its 31 references that are in the map. First posted on arXiv 18 Jan 2022.

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1 of 31 references.

References · 1

  1. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 6; 2026: 8.

19 papers.

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  1. 2025Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemblecited by 13
  2. 2024Physics-informed kernel learningcited by 11
  3. 2025High precision PINNs in unbounded domains: application to singularity formulation in PDEscited by 6
  4. 2026A Machine Learning Approach to the Nirenberg Problemcited by 5
  5. 2026Minimising Willmore Energy via Neural Flowcited by 4
  6. 2024Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNscited by 3
  7. 2024Spectrum-Informed Multistage Neural Networks: Multiscale Function Approximators of Machine Precisioncited by 2
  8. 2026Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networkscited by 2
  9. 2025Unraveling particle dark matter with Physics-Informed Neural Networkscited by 1
  10. 2026PINNs in More General Geometrycited by 1
  11. 2026Fourier Feature Pyramids for Physics-Informed Neural Networkscited by 1
  12. 2026When Does AI for PDEs Yield Scientific Evidence?cited by 1
7 more
  1. 2024Learning Traveling Solitary Waves Using Separable Gaussian Neural Networkscited by 0
  2. 2024Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditionscited by 0
  3. 2025Automated discovery of finite volume schemes using Graph Neural Networkscited by 0
  4. 2025Auto-Adaptive PINNs with Applications to Phase Transitionscited by 0
  5. 2025Resolving Sharp Gradients of Unstable Singularities to Machine Precision via Neural Networkscited by 0
  6. 2026Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networkscited by 0
  7. 2026SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learningcited by 0