Anomalous Gradients in AI: Multivariate Fractional Calculus Unifying Landau Inequalities and Deep Operator Stability

Recent advancements in fractional calculus have revitalized the study of Landau inequalities, particularly in multivariate settings. This work bridges critical gaps in the theory of fractional Landau inequalities by addressing optimal constants, function space geometry, and applications to neural operators. Building on Anastassiou's framework (2025) for directional fractional derivatives in \(\mathbb{R}^k\), we refine fractional Taylor remainder estimates using higher-order asymptotics, yielding sharper gradient bounds for functions in \(W^{\nu,\infty}(\mathbb{R}^k_+)\) with fractional orders \(\nu \in (2,4)\). Our analysis extends classical inequalities to fractional Sobolev spaces \(W^{\nu,p}(\mathbb{R}^k_+)\) via embedding theorems and duality, while variational optimization techniques reveal dimension-dependent constants that tighten existing bounds. For \(\nu \in (2,3)\), we establish \(\| \nabla f \|_\infty \leq 2\sqrt{2}k \cdot \sqrt{\|f\|_\infty K_\nu / \Gamma(\nu+1)}\), where \(K_\nu\) is a novel fractional curvature modulus. The case \(\nu \in (3,4)\) introduces a third-order fractional torsion modulus \(M_\nu\), leading to gradient bounds scaling as \(\|f\|_\infty^{2/3} M_\nu^{1/3}\). Furthermore, we generalize these results to deep neural operators with spectral norm constraints, demonstrating stability under input perturbations via fractional smoothness moduli. Key innovations include the synthesis of multivariate fractional calculus with neural network architecture, yielding layer-dependent bounds for \(\tanh\)-activated networks and mollified ReLU residual networks. Applications to fractional PDE regularity and operator learning are discussed, with numerical implications for training stability and anomaly detection. Our findings unify classical gradient bounds with non-local fractional dynamics, offering a framework for analyzing high-dimensional systems governed by anomalous diffusion or rough geometries.