Abstract
<title>Abstract</title> <p>Turbulence remains one of the last unsolved problems in classical physics, with theoretical descriptions historically fragmented across distinct physical domains—incompressible Navier–Stokes, quantum turbulence, compressible flow, magnetohydrodynamics (MHD), stratified flows, and active matter. We present a unified functional-renormalization-group-assisted Fourier neural operator (FNO×RG) framework that bridges spectral learning and Wilsonian coarse-graining in a three-stage pipeline: (i) learn the spectral closure Γ_κ via FNO from DNS/experimental data; (ii) embed Γ_κ into the Wetterich exact RG equation; (iii) extract fixed points and anomalous dimensions via eigenvalue analysis of the linearized RG flow. Applying this pipeline to six physically distinct systems, our analysis is consistent with known universality classes and suggests the possible existence of multi-competing-fixed-point structures in five of the six cases. Specifically: (1) Navier–Stokes turbulence yields the conditional result η_ν = 4/3 (from the Galilean Ward identity z=2−η_ν with z=2/3), the She–Leveque scaling formula, a two-loop β-function, and consistent results across 18 independent tests (3 confirmed novel predictions, 8 reproduce known results, 0 negated); (2) quantum turbulence reveals a polarization-controlled crossover between Kozik–Svistunov (k^(−7/5)) and L'vov–Nazarenko (k^(−5/3)) spectra, where vortex-line polarization P acts as the RG-relevant operator; our predictions for quantum turbulence await experimental verification; (3) compressible turbulence produces a Mach-dependent β(Ma) = −(5/3 + 2αMa²)/(1+αMa²); (4) MHD turbulence exhibits a double-inertial IK→K41 cascade confirmed by satellite data, with helicity barrier effects; (5) stratified turbulence reveals a triple-fixed-point structure (K41/Bolgov–Obukhov/two-dimensional), with buoyancy scaling dimension [g_b]=−1; (6) active matter turbulence resolves the Nematic-Active Turbulent Fixed Point (k^(−1)) versus Pressure-Active Turbulent Fixed Point (k^(−8/3)) controversy. We identify cross-system universality patterns and discuss limitations including FNO training data requirements and higher-loop corrections.</p>