Interpretable Physics Extraction from Data for Linear Dynamical Systems using Lie Generator Networks

2026-03-28

AI Summary

Existing neural approaches to identifying linear dynamical systems often struggle to simultaneously ensure physical consistency and interpretability, frequently yielding unstable or unphysical solutions. This work proposes Lie Generator Networks (LGN), which enforce intrinsic stability and dissipativity by parameterizing the generator matrix in the form \( A = S - D \), where \( S \) is skew-symmetric and \( D \) is positive diagonal. By leveraging the matrix exponential for exact trajectory computation, LGN circumvents numerical integration errors. Integrating Lie algebraic structure, spectral analysis, and structured neural networks, the method accurately recovers all eigenvalues in a 100-dimensional RLC ladder circuit, achieving mean errors over two orders of magnitude lower than unconstrained baselines. Furthermore, LGN enables direct extraction of key physical quantities such as poles, natural frequencies, and damping ratios.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation

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

When the system is linear, why should learning be nonlinear? Linear dynamical systems, the analytical backbone of control theory, signal processing and circuit analysis, have exact closed-form solutions via the state transition matrix. Yet when system parameters must be inferred from data, recent neural approaches offer flexibility at the cost of physical guarantees: Neural ODEs provide flexible trajectory approximation but may violate physical invariants, while energy preserving architectures do not natively represent dissipation essential to real-world systems. We introduce Lie Generator Networks (LGN), which learn a structured generator A and compute trajectories directly via matrix exponentiation. This shift from integration to exponentiation preserves structure by construction. By parameterizing A = S - D (skew-symmetric minus positive diagonal), stability and dissipation emerge from the underlying architecture and are not introduced during training via the loss function. LGN provides a unified framework for linear conservative, dissipative, and time-varying systems. On a 100-dimensional stable RLC ladder, standard derivative-based least-squares system identification can yield unstable eigenvalues. The unconstrained LGN yields stable but physically incorrect spectra, whereas LGN-SD recovers all 100 eigenvalues with over two orders of magnitude lower mean eigenvalue error than unconstrained alternatives. Critically, these eigenvalues reveal poles, natural frequencies, and damping ratios which are interpretable physics that black-box networks do not provide.

Problem

• linear dynamical systems
• interpretable physics
• system identification
• stability
• dissipation

Innovation

• Lie Generator Networks
• linear dynamical systems
• matrix exponentiation
• structured parameterization
• interpretable physics