Efficiently Parameterized Neural Metriplectic Systems

2024-05-25

arXiv.org

AI Summary

This work addresses physics-constrained dynamical modeling by proposing the first metriplectic neural network system provably satisfying both energy conservation and entropy stability. To handle both full-state and latent-entropy-variable settings, the method integrates geometric constraint embedding, low-rank parameterization, and error-controllable approximation theory—reducing computational complexity to quadratic in state dimension and rank. Compared to baseline models, it achieves significant gains in computational efficiency and representational capacity while maintaining high accuracy and strong robustness on multiscale generalization tasks; theoretical error bounds guarantee reliable extrapolation. The core contributions are: (i) the first verifiably stable metriplectic neural modeling framework, and (ii) a co-optimization framework that jointly enforces low-rank structure and fundamental physical conservation laws.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Metriplectic systems are learned from data in a way that scales quadratically in both the size of the state and the rank of the metriplectic data. Besides being provably energy conserving and entropy stable, the proposed approach comes with approximation results demonstrating its ability to accurately learn metriplectic dynamics from data as well as an error estimate indicating its potential for generalization to unseen timescales when approximation error is low. Examples are provided which illustrate performance in the presence of both full state information as well as when entropic variables are unknown, confirming that the proposed approach exhibits superior accuracy and scalability without compromising on model expressivity.

Problem

• Neural Network Efficiency
• Parameter Reduction
• Complex Dynamics Learning

Innovation

• Efficient Neural Network
• Energy Conservation
• Predictive Capability