Lyapunov–Schmidt Reduction for Fractional Neural Operators for High-Mach Compressible Rotational Flows

Compressible rotational gas–particle flows at high Mach numbers are ubiquitous in advanced powder processing technologies, including cyclone separators, supersonic jet mills, and pneumatic conveying systems. Accurate predictive modelling remains profoundly challenging due to long-range nonlocal particle interactions, anomalous diffusion and viscoelastic memory effects, and shock discontinuities that render classical local models inadequate. To address these challenges, we develop a rigorous mathematical framework that replaces the classical Laplacian with a nonlocal integral operator constructed from symmetrized neural kernels, coupled with a Caputo fractional time derivative of order β∈(0,1) to model subdiffusive transport and rheological memory. A generalised Voronovskaya-type theorem for neural kernel operators furnishes sharp pointwise error bounds and convergence rates even in the presence of discontinuities, rigorously justifying the neural operator approximation. Employing a Lyapunov-Schmidt reduction adapted to this fractional nonlocal setting, we establish the existence, uniqueness, and linear stability of multi-bubble solutions representing interacting coherent structures. The asymptotic expansion of the reduced energy functional yields a novel scaling law λm∼Csm1/s, where s∈(1/2,1) is the fractional exponent—fundamentally different from the classical local case s=1, where λm∼Cm. The framework is validated against experimental and LES data for a Stairmand cyclone separator, achieving RMSE values of 12.39–19.32% across Mach numbers M=1.0,2.0,5.0. The fractional model outperforms classical semi-empirical correlations by up to 70% in predictive accuracy. This work bridges advanced functional analysis with engineering practice, providing a solid foundation for reliable simulations and design optimisation of powder processing equipment, and paving the way for physics-informed neural operator architectures with guaranteed stability and convergence in industrial compressible multiphase flow applications.