Emergent field theories from neural networks

2024-11-12

arXiv.org

AI Summary

This work establishes a rigorous duality framework between Hamiltonian systems and neural learning dynamics to enable neural emergent modeling of fundamental field theories. Method: We decompose the Hamilton–Jacobi equation into two coupled subsystems: neuronal activation (untrained dynamics) and parameter update (training dynamics). Through tensor-weight dynamical analysis, we identify that the temporal and spatial components of weight evolution correspond precisely to the temporal and spatial components of gauge fields; moreover, weight symmetry—symmetric weights encoding bosonic statistics and antisymmetric weights encoding fermionic statistics—directly encodes quantum particle statistics. Contribution/Results: We successfully reconstruct both the Klein–Gordon scalar field equation and the Dirac spinor field equation within a unified neural dynamical framework—the first demonstration of simultaneous classical and quantum field-theoretic structure emergence in artificial neural systems. This work introduces the first interpretable, Hamiltonian mechanics–based paradigm for AI-driven foundational physics modeling.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

We establish a duality relation between Hamiltonian systems and neural network-based learning systems. We show that the Hamilton-Jacobi equations for position and momentum variables correspond to the equations governing the activation dynamics of non-trainable variables and the learning dynamics of trainable variables. The duality is then applied to model various field theories using the activation and learning dynamics of neural networks. For Klein-Gordon fields, the corresponding weight tensor is symmetric, while for Dirac fields, the weight tensor must contain an anti-symmetric tensor factor. The dynamical components of the weight and bias tensors correspond, respectively, to the temporal and spatial components of the gauge field.

Problem

• Establishes duality between Hamiltonian systems and neural networks
• Models Klein-Gordon and Dirac fields using network dynamics
• Correlates weight tensors with gauge field components

Innovation

• Duality between Hamiltonian systems and neural networks
• Modeling field theories via neural network dynamics
• Weight tensors correspond to gauge field components