Artificial Neural Networks on Graded Vector Spaces

TL;DR

This paper introduces a novel neural network framework that operates over graded vector spaces, enabling modeling of hierarchical and structured data with applications in algebraic geometry and physics.

Contribution

It develops graded neural architectures and loss functions grounded in algebraic theory, extending classical neural networks to handle graded structures and symmetries.

Findings

Outperforms standard neural networks in predicting invariants in weighted projective spaces.

Effectively models supersymmetric systems with graded neural networks.

Demonstrates practical utility through case studies.

Abstract

This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic properties of graded vector spaces, where features carry distinct weights, we extend classical neural networks with graded neurons, layers, and activation functions that preserve structural integrity. Grounded in group actions, representation theory, and graded algebra, our approach combines theoretical rigor with practical utility. We introduce graded neural architectures, loss functions prioritizing graded components, and equivariant extensions adaptable to diverse gradings. Case studies validate the framework's effectiveness, outperforming standard neural networks in tasks such as predicting invariants in weighted projective spaces and modeling supersymmetric systems. This work establishes a new frontier in machine learning, merging mathematical sophistication with interdisciplinary applications. Future challenges, including computational scalability and finite field extensions, offer rich opportunities for advancing this paradigm.