Context-fNODE: Context-aware Fractional Neural Ordinary Differential Equation

While standard Neural Ordinary Differential Equations (ODEs) excel at modeling continuous dynamics, their strictly Markovian nature limits their capacity to capture the macroscopic long-tail memory inherent in physical kinematics. Existing fractional neural models address this historical inertia but restrictively assume isotropic scalar operators and fully autonomous, closed environments. To overcome these limitations, we propose the Context-aware Fractional Neural ODE (Context-fNODE), a unified mixed-order architecture. Context-fNODE replaces commensurate scalar operators with an incommensurate, vectorized Grünwald-Letnikov derivative to natively capture direction-specific anisotropic inertia. This framework is mathematically coupled with a co-evolving first-order latent environment to absorb unobservable exogenous perturbations. Theoretically, we prove this formulation bounds the spatial error manifold, strictly suppresses chaotic Lyapunov divergence, and preserves a linear numerical convergence rate. Extensive evaluations on chaotic dynamic systems and complex 3D kinematics demonstrate that Context-fNODE significantly outperforms state-of-the-art baselines, establishing superior long-horizon forecasting endurance in non-autonomous, anisotropic environments.