Graded Neural Networks

We introduce a rigorous framework for Graded Neural Networks, a new class of architectures built on coordinate-wise graded vector spaces [Formula: see text]. Using an algebraic scalar action [Formula: see text] defined by a grading tuple [Formula: see text], we construct grade-sensitive neurons, activations, and loss functions that embed hierarchical feature structure directly into the network’s architecture. This grading endows GNNs with enhanced expressivity and interpretability, extending classical neural networks as a special case. We develop both the algebraic theory and computational implementation of GNNs, addressing challenges such as numerical instability and optimization with anisotropic scaling. Theoretical results establish universal approximation for graded-homogeneous functions, along with convergence rates in graded Sobolev and Besov spaces. We also show that GNNs achieve lower complexity for approximating structured functions compared to standard networks. Applications span hierarchical data modeling, quantum systems, and photonic hardware, where grades correspond to physical parameters. This work provides a principled foundation for incorporating grading into neural computation, unifying algebraic structure with learning, and opening new directions in both theory and practice.