先前在简化方程中找到的多数奇异性是「稳定」的，意思是在小幅改动下仍会持续存在；但许多人预期任何符合现实理论的奇异性会是「不稳定」的，只在极其精确的初始条件下才会发生。一个团队在 2025 年 9 月发布的预印本（arXiv:2509.14185）中报告，将机器学习方法训练用于搜寻已知会产生奇异性的模型，找到了额外的潜在 blowup 情境，包括不稳定候选者；他们也在其他流体方程中发现更多不稳定候选者，这标志著首次在具有超过 1 个空间维度的流体模型中辨识出不稳定奇异性候选者（但并非在 Navier-Stokes 本身）。
文章将此与先前在无黏性 Euler 方程中的计算式奇异性搜寻作对比：2013 年，Thomas Hou 与 Guo Luo 模拟一个「can」配置，令两半部反向旋转，并在边界附近看到涡度（vorticity）增长到超出数值极限；2022 年，Hou 与 Jiajie Chen 使用电脑辅助方法，将该候选者转化为真正奇异性的证明，时间线跨越近 1 decade。不稳定候选者仍更难，因为有限精度的模拟误差，甚至极微小的扰动，都可能消除 blowup；因此新结果带来乐观，显示不稳定性不必然使候选者无法被发现，但仍需要严格证明新找到的候选者确实会 blow up，并指出他们尚未产生任何 $1 million 的 Navier-Stokes 奇异性。
The Navier-Stokes equations, finalized nearly 200 years ago by Claude-Louis Navier and George Gabriel Stokes, underpin modern fluid modeling from ocean currents to airflow around wings, yet mathematicians suspect rare “blowups” where a quantity becomes infinite and the predicted motion turns unphysical. Proving a blowup exists in 3D Navier-Stokes or proving it never happens is a Clay Millennium Problem with a $1 million prize, so researchers often probe simplified fluid equations (including 1D models) as stepping stones toward the 3D case.
Most previously found singularities in simplified equations were “stable,” meaning they persist under small changes, but many expect any realistic-theory singularities to be “unstable,” occurring only for extremely precise initial conditions. A team reported in a preprint posted in September 2025 (arXiv:2509.14185) that training machine-learning methods to search known singularity-bearing models uncovered additional potential blowup scenarios, including unstable candidates; they also found more unstable candidates in other fluid equations, marking the first time an unstable singularity candidate was identified in a fluid model with more than 1 spatial dimension (though not in Navier-Stokes itself).
The article contrasts this with earlier computational singularity hunting in the inviscid Euler equations: in 2013, Thomas Hou and Guo Luo simulated a “can” setup with counter-rotating halves and saw vorticity grow beyond numerical limits near the boundary, and in 2022 Hou and Jiajie Chen used computer-assisted methods to turn that candidate into a proof of a true singularity, a timeline spanning nearly 1 decade. Unstable candidates remain harder because finite-precision simulation errors and even tiny perturbations can eliminate the blowup, so the new results raise optimism by showing that instability need not make candidates undiscoverable, while still requiring rigorous proofs that the newly found candidates actually blow up and noting they have not yet produced any $1 million Navier-Stokes singularity.