Dirac-Interconnected Neural Elements: Discovering Modularity in Physical Systems Without Reduction

2026-10-02

AI Summary

This study addresses the limitations of existing physical system modeling approaches, which often struggle to fully exploit component interconnection properties and rely heavily on prior knowledge or dimensionality reduction assumptions. We propose DINEs, a model that represents systems as differential-algebraic equations subject to Dirac structure constraints, leveraging deep learning to jointly characterize component dynamics and interconnection topologies. This formulation preserves unreduced representations, accommodates partially observable scenarios, and enables flexible isolation and composition of subsystems without retraining. Experimental results demonstrate that DINEs significantly outperforms existing methods in module discovery and efficient reconstruction tasks for complex physical systems.

Abstract

Deep learning has shown remarkable success in the data-driven modeling of dynamical systems. Much of its success is attributed not to the flexibility of neural networks but to inductive biases based on physical prior knowledge, such as energy conservation and symplecticity. However, existing methods do not fully exploit the fact that real-world physical systems are interconnections of components. Some methods require the interconnection to be known a priori, while others assume the system to be reducible to an ordinary differential equation (ODE) and learn only the reduced ODE, discarding the algebraic constraints imposed by the interconnection. Here, we propose Dirac-interconnected neural elements (DINEs), a neural network model that represents a physical system as a differential-algebraic equation (DAE), whose algebraic constraints are given by a Dirac structure in kernel representation. With DINEs, we simultaneously identify from data the interconnection among the components as a Dirac structure and learn the characteristics of the components as neural networks. This allows us to keep the learned subsystems in unreduced form and isolate or compose them to make a new system without retraining. Moreover, DINEs can handle partially observable systems. Experimental results demonstrate these capabilities on physical systems beyond the reach of existing methods.

Problem

• dynamical systems modeling
• Dirac structure
• differential-algebraic equations
• modularity
• partially observable systems

Innovation

• Dirac-interconnected neural elements
• differential-algebraic equation
• modularity
• physical systems
• partially observable systems