SIREN Residual Error as a Regularity Diagnostic for Navier-Stokes Equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910058812538880 |
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| author | Burton, Jason |
| author_facet | Burton, Jason |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18061 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | SIREN Residual Error as a Regularity Diagnostic for Navier-Stokes Equations Burton, Jason Fluid Dynamics Analysis of PDEs 35Q30, 65M99 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. |
| title | SIREN Residual Error as a Regularity Diagnostic for Navier-Stokes Equations |
| topic | Fluid Dynamics Analysis of PDEs 35Q30, 65M99 |
| url | https://arxiv.org/abs/2603.18061 |