New families of unstable singularities in fluid equations
Physics-informed neural networks found previously unknown unstable self-similar blow-up solutions for three fluid equations, computed to near machine precision, and revealed an unexpected near-linear relation between a solution's instability order and its blow-up rate.
- Model
- Physics-informed neural networks with Gauss–Newton optimisation
- Field
- Mathematics
- Date
- 2025-09-17
- Human collaborators
- Javier Gómez-Serrano, Tristan Buckmaster, Ching-Yao Lai, Yongji Wang
Sources
Original work
Announcement
Media coverage
What was found
A physics-informed neural network minimises the residual of the differential equation itself rather than fitting data, which lets it hunt for self-similar blow-up profiles that ordinary simulation cannot hold onto because they are repelling. The team reports a stable and three unstable singularities for the incompressible porous media equation, a stable and three unstable solutions plus a candidate fourth for 2D Boussinesq, and an unstable singularity for 3D Euler with boundary. Residuals were driven near machine precision, the accuracy level a later computer-assisted proof would need. Across equations the blow-up rate depends almost linearly on the order of instability, a pattern nobody had predicted.
Novelty check
Stable self-similar blow-up for these equations was already known, including Chen and Hou's computer-assisted proof for 3D Euler with boundary. Unstable singularities had been found only in isolated cases because they are numerically repelling; no systematic families existed in the literature. This is the first paper to produce several unstable families across three different equations by one method.
Caveats and known objections
An arXiv preprint, not peer reviewed at the time of this entry. These are high-precision numerical solutions, not theorems: converting them into proofs requires a computer-assisted verification step the paper leaves to future work. No singularity for 3D Navier–Stokes, the Millennium Prize problem, is claimed, and the equations studied are not that one. Autonomy is search-scaffold: the equations, the self-similar ansatz and the loss function are human-designed and the network is the solver.
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Cite this entry
whataifound.org. (2025). New families of unstable singularities in fluid equations. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2025-09-17-unstable-singularitiesBibTeX
@misc{whataifound-googledeepmind-2025-singularities,
title = {New families of unstable singularities in fluid equations},
author = {{whataifound.org}},
year = {2025},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Google DeepMind (with Brown, NYU and Stanford). Verification: Author verified. Autonomy: Search scaffold.},
url = {https://whataifound.org/finding/2025-09-17-unstable-singularities}
}