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SIREN Residual Error as a Regularity Diagnostic for Navier-Stokes Equations

Jason Burton

Abstract

We introduce a method for detecting regularity loss in solutions to the three-dimensional Navier-Stokes equations using the approximation error of Sinusoidal Representation Networks (SIRENs). SIRENs use sin() activations, producing C-infinity outputs that cannot represent non-smooth features. By classical spectral approximation theory, the SIREN error is bounded by O(N^{-s}) where s is the local Sobolev regularity. At a singularity (s to 0), the error is O(1) and localizes via the Gibbs phenomenon. We decompose the velocity field into a cheap analytical baseline (advection-diffusion) and a learned residual (pressure correction), training a compact SIREN (4,867 parameters). We validate on the 3D Taylor-Green vortex, where error concentration increases from 4.9x to 13.6x as viscosity decreases from 0.01 to 0.0001, localizing to the stagnation point -- the geometry matching the singularity proven by Chen and Hou (2025) for 3D Euler. On axisymmetric equations, we reproduce blowup signatures (T* converging across resolutions) and identify a critical viscosity nu_c = 0.00582 for the regularization transition.

SIREN Residual Error as a Regularity Diagnostic for Navier-Stokes Equations

Abstract

We introduce a method for detecting regularity loss in solutions to the three-dimensional Navier-Stokes equations using the approximation error of Sinusoidal Representation Networks (SIRENs). SIRENs use sin() activations, producing C-infinity outputs that cannot represent non-smooth features. By classical spectral approximation theory, the SIREN error is bounded by O(N^{-s}) where s is the local Sobolev regularity. At a singularity (s to 0), the error is O(1) and localizes via the Gibbs phenomenon. We decompose the velocity field into a cheap analytical baseline (advection-diffusion) and a learned residual (pressure correction), training a compact SIREN (4,867 parameters). We validate on the 3D Taylor-Green vortex, where error concentration increases from 4.9x to 13.6x as viscosity decreases from 0.01 to 0.0001, localizing to the stagnation point -- the geometry matching the singularity proven by Chen and Hou (2025) for 3D Euler. On axisymmetric equations, we reproduce blowup signatures (T* converging across resolutions) and identify a critical viscosity nu_c = 0.00582 for the regularization transition.
Paper Structure (33 sections, 6 equations, 5 figures, 2 tables)

This paper contains 33 sections, 6 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Background
  4. Navier-Stokes Equations
  5. Axisymmetric Formulation
  6. Related Work
  7. PINN failure modes.
  8. Neural operator error bounds.
  9. Spectral bias.
  10. AMR.
  11. SIREN Architecture and Spectral Approximation Theory
  12. Method
  13. Residual Decomposition
  14. SIREN Training
  15. Regularity Diagnostic
  16. ...and 18 more sections

Figures (5)

  • Figure 1: Residual decomposition pipeline. The full NS solution decomposes into an advection-diffusion baseline and a pressure correction residual. A compact SIREN (4,867 parameters) learns the residual; its approximation error field $\varepsilon(\mathbf{x})$ serves as the regularity diagnostic.
  • Figure 2: Error concentration vs viscosity. Left: concentration ratios at $t=0$ and $t=T$ for three viscosities. Right: schematic 1D error profiles showing progressive localization at the stagnation point $(\pi,\pi,\pi)$ as $\nu$ decreases.
  • Figure 3: Blowup time convergence. Left: $1/\|\omega\|_\infty$ vs time, showing linear decrease toward $T^* \approx 0.74$ for Euler at both resolutions. NS with $\nu=0.01$ shows damped growth; $\nu=0.001$ matches Euler. Right: $\|\omega\|_\infty$ on log scale.
  • Figure 4: Critical viscosity transition. Left: vorticity growth slope $a$ vs viscosity $\nu$, showing regularized (circles) and Euler-like (diamonds) behavior separated by the threshold $a=-5.0$. Right: zoomed view of the transition region $\nu_c = 0.00582 \pm 0.00004$, spanning $\Delta\nu = 0.00007$.
  • Figure 5: SIREN corrector architecture. 7 inputs $\to$ 64-unit hidden layers with $\sin(\omega_0 \cdot)$ activation $\to$ 3 velocity correction outputs. Total: 4,867 parameters.