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

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.