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

arXiv: https://arxiv.org/abs/2603.27442 · HTML: https://mlsanity.com/paper/2603.27442/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 21 references that are in the map.
- First posted on arXiv 28 Mar 2026.

## Builds on (most cited first, 13 of 13)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Inverse Problems for Matrix Exponential in System Identification: System Aliasing](https://mlsanity.com/paper/1605.06973/) (2016)

## Built on by (most cited first, 1 of 1)

- [Lie Generator Networks for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2603.29264/) (2026)

## Abstract (arXiv)

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.