Greg Brockman 引用 @scottnarmstrong 的评论称,OpenAI 这篇论文在技术层面令人难以置信,可能超出人类能想到的范围,但仍在人类理解之内。被引评论认为该构造与凸积分(convex integration)关系更大,其做法是让流体不稳定性自行产生振荡、由雷诺应力形成使光滑涡坍缩的力,这种组合在既有文献中没有先例;已有多个 PDE 专家团队表示大致读懂了论文,预计很快会有解读。该评论还称,认为这一结果并不令人意外的人是在胡说。
引用数学界人士对 OpenAI 一篇偏微分方程论文的技术拆解,呈现该结果与凸积分方法的关联及学界反应。
“从技术层面来看,这篇论文令人难以置信,或许超出了人类所能想到的范围,但并未超出人类的理解范围。”
由于我现在被 bluey 屏蔽,无法回复这个帖子,我在这里通过引用推文来回答 OpenAI 的构造与凸积分的关系比其他任何东西都更密切。 在凸积分中,振荡通常是手工构造的,而迭代过程会产生狂野的弱解。Albritton–Brué–Colombo 的非唯一性构造利用了一个不稳定的涡旋来将解分裂开来。Córdoba–Martínez-Zoroa 奇点通过应变在各个尺度上放大涡量。在 OpenAI 的论文中,流体不稳定性被巧妙地设计成让流动本身产生振荡:它们通过从背景剪切中汲取能量来增长。雷诺应力随后成为产生坍缩的平滑涡旋的力。以这种方式利用物理不稳定性来产生应力是前所未有的,而正是这一点产生了平滑的力,而非仅仅是粗糙的力。 其中一些想法在文献中已有(例如 Leibovich-Stewartson 1983),但它们组合的方式绝对是疯狂的,而且绝对在之前的文献中从未出现过。 我也不知道有任何人曾提出凸积分会在这样的爆破解的构造中发挥作用。 在技术层面上,这篇论文令人难以置信,也许超出了人类所能想到的范围,但并未超出人类所能理解的范围:有几个不同的 PDE 专家组正在努力理解它,他们说他们现在或多或少已经理解了。我预计很快就会被消化吸收 td;lr:任何说这个 OAI 结果不足为奇的人都是在胡说八道 抱歉!
原文
Since I'm now blocked by bluey from replying to this thread, I answer here by quote tweet the openAI construction has more to do with convex integration than anything else. In convex integration, the oscillations are typically built by hand and the iteration procedure creates wild weak solutions. The non-uniqueness construction of Albritton–Brué–Colombo used an unstable vortex to split solutions apart. The Córdoba–Martínez-Zoroa singularities amplify vorticity across scales by strain. In the openAI paper, a fluid instability is clever made so that the flow itself generates the oscillations: they grow by drawing energy from the background shear. The Reynolds stress then becomes the force that creates the collapsing smooth vortex. Using a physical instability to create the stress in this way has no precedent, and it is what produces a smooth force rather than a merely rough one. Some on these ideas are in the literature (e.g. Leibovich-Stewartson 1983) but the way they are combined is definitely insane and definitely nowhere in the literature previously. I also don't know of anyone suggesting that convex integration would play a role in the creation of a blow-up solution like this. On a technical level, the paper is unbelievable, maybe beyond what a human could have come up with but not beyond human understanding: there are several different groups of pde experts working on understanding it who say that they more or less understand now. I expect digestions soon td;lr: anyone saying that this OAI result is unsurprising is bullshitting sorry!
来源:Greg Brockman · x.com