# Lie Generator Networks for Nonlinear Partial Differential Equations

arXiv: https://arxiv.org/abs/2603.29264 · HTML: https://mlsanity.com/paper/2603.29264/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 15 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [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)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Towards Efficient and Stable Ocean State Forecasting: A Continuous-Time Koopman Approach](https://mlsanity.com/paper/2603.05560/) (2026)
- [Interpretable Physics Extraction from Data for Linear Dynamical Systems using Lie Generator Networks](https://mlsanity.com/paper/2603.27442/) (2026)

## Abstract (arXiv)

Linear dynamical systems are fully characterized by their eigenspectra, accessible directly from the generator of the dynamics. For nonlinear systems governed by partial differential equations, no equivalent theory exists. We introduce Lie Generator Network-Koopman (LGN-KM), a neural operator that lifts nonlinear dynamics into a linear latent space and learns the continuous-time Koopman generator (Lk) through a decomposition Lk = S - Dk, where S is skew-symmetric representing conservative inter-modal coupling, and Dk is a positive-definite diagonal encoding modal dissipation. This architectural decomposition enforces stability and enables interpretability through direct spectral access to the learned dynamics. On two-dimensional Navier--Stokes turbulence, the generator recovers the known dissipation scaling and a complete multi-branch dispersion relation from trajectory data alone with no physics supervision. Independently trained models at different flow regimes recover matched gauge-invariant spectral structure, exposing a gauge freedom in the Koopman lifting. Because the generator is provably stable, it enables guaranteed long-horizon stability, continuous-time evaluation at arbitrary time, and physics-informed cross-viscosity model transfer.