Twisted Multiparameter Neural Operators for Anisotropic Wave Dynamics

Continuous neural operators have emerged as a prominent paradigm for learning resolution-independent mappings between infinite-dimensional function spaces in partial differential equations (PDEs). However, conventional spectral operators rely on Cartesian tensor-product multipliers that are inherently smooth across diagonal frequency rays, fundamentally limiting their capacity to represent oblique shocks, tilted shear interfaces, and anisotropic wave-fronts without severe parameter explosion. In this work, we introduce the Twisted Multiparameter Neural Operator (TMNO), a mathematically principled operator learning framework grounded in the harmonic analysis of twisted multiparameter singular integrals. By pushing forward a multi-parameter product structure through an algebraic quotient projection, TMNO parameterizes non-tensor singular multipliers concentrated along the coupled singular variety ${\Sigma }^{\vee }=\{{\xi }_1=0\}\cup \{{\xi }_2=0\}\cup \{{\xi }_1+{\xi }_2=0\}$, unlocking a native tri-axial Calder\'on multiscale decomposition on two-dimensional domains. To physically align with directional transport without grid-interpolation dissipation or frequency aliasing, TMNO incorporates an automorphic tube branch governed by exact ${\mathrm{SL}}_2(\mathbb{Z})$ modular lattice shears, while structurally decoupling representational parameter capacity from computational FLOP throughput. Across diverse fluid convection, high-contrast acoustic scattering, and compressible shock reflection benchmarks, TMNO demonstrates superior multiscale front resolution, robust out-of-distribution generalization under rotated shock profiles, and orders-of-magnitude higher parameter efficiency.