A SIREN — a neural network with sinusoidal activations — produces infinitely differentiable outputs. It cannot represent a function with a singularity. Give it one, and it fails. The failure concentrates exactly where the singularity lives.
Burton noticed this and turned it around. Instead of asking how well the network approximates the Navier-Stokes velocity field, ask where it fails. The residual error doesn't scatter randomly. It concentrates near the stagnation point of a 3D Taylor-Green vortex, intensifying from 4.9x to 13.6x background as viscosity drops from 0.01 to 0.0001. The concentration geometrically matches where Chen and Hou placed singularities for 3D Euler equations.
The method uses 4,867 parameters. Not millions. Not even tens of thousands. A compact smooth approximator, deployed against a potentially singular flow, and the failure signature alone identifies the critical viscosity threshold at 0.00582 — the boundary between regularized and singular behavior.
The structural insight: incompetence, when its nature is understood, becomes an instrument. The SIREN cannot represent non-smooth features because its sinusoidal basis is infinitely differentiable. This isn't a bug to fix but a constraint to exploit. Spectral approximation theory gives the bound: error scales as O(N^{-s}) where s is local Sobolev regularity. As s approaches zero — as smoothness disappears — the error approaches O(1) and concentrates via Gibbs phenomena. The failure mode has a known mathematical structure, and that structure is informative.
This inverts the usual relationship between a tool and its limitations. Normally you want tools that succeed everywhere. Here, you want a tool that succeeds everywhere except where the interesting physics lives, and whose failure is diagnostic rather than merely uninformative. The SIREN doesn't find singularities by detecting them. It finds them by being constitutionally unable to accommodate them.
The approach suggests a general strategy: match the representation capacity of your approximator to the regularity of your target, and read the residual as data about what was left out. The approximation error is not noise. It is a measurement of the gap between what the tool can express and what the system demands — and that gap, properly characterized, tells you something the successful approximation cannot.