The Navier-Stokes equations describe how fluids flow. Whether their solutions stay smooth or can develop singularities — points where velocity or pressure becomes infinite — is one of the seven Millennium Prize Problems. A million dollars awaits anyone who proves smoothness or constructs a singularity. After nearly a century, no one has done either.
Part of the difficulty is computational. Traditional numerical simulations solve the equations step by step, marching forward in time, computing each moment from the last. This works for smooth flows. It fails for singularities. A singularity, by definition, is a point where the solution diverges, and the instability of the divergence infects the simulation itself. Small numerical errors — unavoidable in floating-point arithmetic — get amplified by the same forces that drive the singularity. The simulation participates in the instability it's trying to observe. The closer you approach the singularity, the more the computational errors grow, until the method collapses before the physics does.
Unstable singularities are worse. A stable singularity attracts nearby solutions — perturbations decay, and the blow-up proceeds regardless of small errors. An unstable singularity repels nearby solutions — any perturbation, however small, pushes the system away from the singular trajectory. Numerical simulation, which introduces perturbations at every step, cannot stay on an unstable trajectory. It is expelled by the very instability it seeks.
Buckmaster, Lai, and collaborators at Google DeepMind (2025-2026) found unstable singularities using physics-informed neural networks (PINNs). Their method does not simulate the fluid. Instead, it searches directly for self-similar solutions — patterns that look the same at every scale near the blow-up point. The PINN is trained to minimize residuals of the PDE at the self-similar shape, not to evolve the fluid forward in time. It never marches through the instability. It targets the geometry of the singularity from outside.
The result: four new unstable singularities in the Euler equations, four candidates in porous medium equations (one stable, three unstable), and additional unstable singularities in the Córdoba-Córdoba-Fontelos equations. Billion-fold improvement in precision over previous numerical approaches. The unstable singularities had been invisible to every previous method.
The through-claim: the method that does not simulate the process finds the process's most extreme behavior. The traditional simulation fails because it engages with the dynamics — it computes the flow, and the flow's instability destroys the computation. The PINN succeeds because it disengages from the dynamics — it computes the shape of what happens, not the happening itself. It solves for the fixed point of the self-similar transformation, which is a static object, rather than tracking the fluid's evolution toward that object.
The general pattern: when a process is unstable, observing it by participating in it amplifies the instability. The observer inherits the system's fragility. A method that stands outside the process — that characterizes its endpoint without traversing its path — avoids the amplification. The detachment is not a limitation. It is the enabling condition.