From Classical to Quantum Turbulence: A Unified FNO×RG Framework Across Six Physical Systems

Technical Summary: From Classical to Quantum Turbulence: A Unified FNO×RG Framework

Problem Statement

Turbulence remains a fragmented field in physics, lacking a single theoretical framework that unifies its diverse manifestations across distinct physical domains. Historically, descriptions have been siloed: incompressible Navier–Stokes (NS) turbulence relies on Kolmogorov–Obukhov (K41) theory and intermittency corrections (e.g., She–Leveque); quantum turbulence in superfluids involves competing Kozik–Svistunov (KS) and L'vov–Nazarenko (LN) spectral theories; while compressible, magnetohydrodynamic (MHD), stratified, and active matter flows possess their own phenomenological models (e.g., Iroshnikov–Kraichnan, Bolgiano–Obukhov). The renormalization group (RG) approach has provided formal closure mechanisms (e.g., Martin–Siggia–Rose, functional RG), but often relies on perturbative approximations that struggle with non-perturbative physics. Conversely, machine learning approaches like Fourier Neural Operators (FNOs) offer powerful spectral learning capabilities but lack the rigorous theoretical grounding of RG theory to extract universal fixed points and critical exponents.

Methodology: The FNO×RG Three-Stage Pipeline

The paper proposes a unified framework, FNO×RG, which bridges spectral learning and Wilsonian coarse-graining through a three-stage pipeline:

1. Stage I: FNO Spectral Learning

   • A Fourier Neural Operator (FNO) is trained on Direct Numerical Simulation (DNS) or experimental data to learn the scale-dependent spectral closure function, $\Gamma_\kappa$.
   • This function encodes the effective eddy viscosity $\nu_{\text{eff}}(\kappa)$ and the renormalized three-point vertex $V_{\text{eff}}(\kappa_1, \kappa_2, \kappa_3)$.
   • The architecture utilizes 4 Fourier layers with 64 modes per layer and a lifting dimension of 128. The training objective minimizes the spectral closure error derived from the exact relation between the energy spectrum $E(\kappa)$ and the nonlinear energy transfer function $T(\kappa)$.

2. Stage II: Wetterich RG Embedding

   • The learned closure $\Gamma_\kappa$ is embedded into the exact Wetterich functional RG equation for the effective average action $\Gamma_k[\phi]$.
   • Instead of using a perturbative approximation for the second functional derivative $\Gamma_k^{(2)}$, the framework substitutes the FNO-learned non-perturbative corrections: $\Gamma_k^{(2)} \to \Gamma_{k,\text{pert}}^{(2)} + \delta\Gamma_{k,\text{FNO}}^{(2)}$.
   • The Litim regulator is employed to ensure a well-defined fixed-point structure and facilitate analytic extraction of exponents.

3. Stage III: Fixed-Point Extraction

   • The linearized RG flow matrix is analyzed around fixed points to extract critical exponents, anomalous dimensions, and stability properties.
   • The framework identifies fixed points (zeros of $\beta$-functions), relevant/irrelevant directions (eigenvalues of the stability matrix), and crossover trajectories connecting different fixed points.

Key Contributions and Results

The framework was applied to six physically distinct systems, revealing a pattern of multi-competing fixed points in five of the six cases.

1. Incompressible Navier–Stokes Turbulence

• Results: The framework yields a conditional eddy viscosity exponent $\eta_\nu = 4/3$ (derived from the Galilean Ward identity $z = 2 - \eta_\nu$ with $z=2/3$).
• She–Leveque Formula: It derives the She–Leveque scaling formula $\zeta_p = p/9 + 2[1 - (2/3)^{p/3}]$ from first principles without adjustable parameters, identifying dissipation structures as sheet-like (codimension 1).
• Validation: The model was validated against 18 independent tests (DNS, experiments, theory). Results included 3 confirmed novel predictions, 8 reproductions of known results, and 0 negated results.
• Fixed Points: A unique isotropic fixed point was found, with a secondary bifurcation fixed point appearing under anisotropic perturbations.

2. Quantum Turbulence

• Results: The analysis reveals a polarization-controlled crossover between the KS spectrum ($E(k) \propto k^{-7/5}$) and the LN spectrum ($E(k) \propto k^{-5/3}$).
• Mechanism: Vortex-line polarization $P$ acts as an RG-relevant operator. A critical polarization $P_c \approx 0.31$ separates the regimes: high polarization ($P > P_c$) attracts the flow to the KS fixed point, while low polarization ($P \approx 0$) attracts it to the LN fixed point.
• Status: These predictions await experimental verification.

3. Compressible Turbulence

• Results: A Mach-number-dependent $\beta$-function is derived: $\beta(\text{Ma}) = -(5/3 + 2\alpha \text{Ma}^2)/(1 + \alpha \text{Ma}^2)$.
• Crossover: This predicts a continuous transition from the K41 $k^{-5/3}$ spectrum (incompressible limit) to a Burgers-like $k^{-2}$ spectrum (supersonic limit), with a crossover occurring near $\text{Ma} \approx 0.91$.
• Fixed Points: A continuous line of fixed points parameterized by Mach number connects the K41 and Burgers regimes.

4. MHD Turbulence

• Results: The framework confirms a double-inertial subrange structure: an Iroshnikov–Kraichnan (IK) regime ($k^{-3/2}$) at large scales transitioning to a K41 regime ($k^{-5/3}$) at small scales.
• Validation: This structure is consistent with solar wind data from the Cluster and Parker Solar Probe missions.
• Novelty: A "helicity barrier" is identified, where high cross-helicity suppresses the turbulent cascade, explaining observed "Alfvénic" states in the solar wind.

5. Stratified Turbulence

• Results: A triple-fixed-point structure is identified: K41 ($k^{-5/3}$), Bolgiano–Obukhov ($k^{-11/5}$), and a 2D limit ($k^{-3}$).
• Mechanism: The buoyancy coupling has a scaling dimension $[g_b] = -1$, making it irrelevant in the UV (stabilizing K41 at small scales) but relevant in the IR (driving the BO regime at large scales).
• Phase Diagram: The dominant fixed point depends on the Froude number ($Fr$) and buoyancy Reynolds number ($Re_b$).

6. Active Matter Turbulence

• Results: The framework resolves the controversy between the Nematic Active Turbulent Fixed Point (NATFP, $k^{-1}$) and the Pressure Active Turbulent Fixed Point (PATFP, $k^{-8/3}$).
• Mechanism: Both are genuine fixed points. The observed spectrum is determined by the activity–pressure ratio $\Pi_a$. Extensile systems with strong activity favor NATFP, while contractile or pressure-dominated systems favor PATFP.

Cross-System Patterns and Significance

The paper identifies several universal patterns across the six systems:

• Multi-Fixed-Point Principle (MFPP): Five of the six systems exhibit multiple competing fixed points, suggesting that turbulence is not a single phenomenon but a family of dynamical regimes selected by physical control parameters (e.g., Mach number, polarization, Froude number).
• Anomalous Dimension Clustering: The anomalous dimensions $\eta$ across all systems cluster tightly around $\bar{\eta} = 0.039 \pm 0.005$, suggesting a quasi-universal quantity for turbulent systems.
• Stability Hierarchy: Simpler fixed points (closer to Gaussian) are typically UV-stable (small scales), while complex fixed points are IR-stable (large scales).

Significance Claims
The authors claim the FNO×RG framework offers a unified, data-driven, yet theory-constrained approach to turbulence that:

1. Unifies Disparate Domains: It applies a single pipeline to classical, quantum, and active turbulent systems.
2. Resolves Controversies: It provides a mechanism for previously conflicting theories (e.g., KS vs. LN, NATFP vs. PATFP) by identifying them as distinct fixed points in the same RG flow.
3. Derives from First Principles: It derives phenomenological formulas (like She–Leveque) from the underlying equations without ad-hoc assumptions.
4. Predicts Transitions: It predicts qualitative transitions in turbulent behavior based on tunable control parameters.

Limitations
The paper acknowledges several limitations:

• Data Dependence: The FNO requires high-quality DNS/experimental data, which is limited for some systems (e.g., quantum turbulence).
• Finite-Size Effects: DNS resolution limits may contaminate large-scale fixed-point analysis.
• Truncation: The analysis truncates the derivative expansion of the effective action, potentially missing higher-order effects in strongly anisotropic or topological systems.
• Sample Size: Claims of universality are based on six systems and require validation on additional systems (e.g., rotating turbulence, dusty plasmas).

The work concludes that turbulence may comprise distinct dynamical regimes governed by different fixed points, with the observed behavior determined by the RG flow trajectory selected by system-specific parameters.