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

arXiv: https://arxiv.org/abs/2201.06780 · HTML: https://mlsanity.com/paper/2201.06780/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 31 references that are in the map.
- First posted on arXiv 18 Jan 2022.

## Builds on (most cited first, 1 of 1)

- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)

## Built on by (most cited first, 19 of 19)

- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [High precision PINNs in unbounded domains: application to singularity formulation in PDEs](https://mlsanity.com/paper/2506.19243/) (2025)
- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [Minimising Willmore Energy via Neural Flow](https://mlsanity.com/paper/2604.04321/) (2026)
- [Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs](https://mlsanity.com/paper/2404.12282/) (2024)
- [Spectrum-Informed Multistage Neural Networks: Multiscale Function Approximators of Machine Precision](https://mlsanity.com/paper/2407.17213/) (2024)
- [Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks](https://mlsanity.com/paper/2606.20467/) (2026)
- [Unraveling particle dark matter with Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.17597/) (2025)
- [PINNs in More General Geometry](https://mlsanity.com/paper/2604.25020/) (2026)
- [Fourier Feature Pyramids for Physics-Informed Neural Networks](https://mlsanity.com/paper/2605.24278/) (2026)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks](https://mlsanity.com/paper/2403.04883/) (2024)
- [Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions](https://mlsanity.com/paper/2407.20741/) (2024)
- [Automated discovery of finite volume schemes using Graph Neural Networks](https://mlsanity.com/paper/2508.19052/) (2025)
- [Auto-Adaptive PINNs with Applications to Phase Transitions](https://mlsanity.com/paper/2510.23999/) (2025)
- [Resolving Sharp Gradients of Unstable Singularities to Machine Precision via Neural Networks](https://mlsanity.com/paper/2511.22819/) (2025)
- [Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.21988/) (2026)
- [SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning](https://mlsanity.com/paper/2608.29448/) (2026)

## 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.