Lie Generator Networks for Nonlinear Partial Differential Equations
arXiv:2603.29264v1 Announce Type: new Abstract: 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 ($L_k$) through a decomposition $L_k = S - D_k$, where $S$ is skew-symmetric representing conservative inter-modal coupling, and $D_k$ 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
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Abstract: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 ($L_k$) through a decomposition $L_k = S - D_k$, where $S$ is skew-symmetric representing conservative inter-modal coupling, and $D_k$ 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.
Comments: 16 pages, 8 figures
Subjects:
Machine Learning (cs.LG); Fluid Dynamics (physics.flu-dyn)
Cite as: arXiv:2603.29264 [cs.LG]
(or arXiv:2603.29264v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2603.29264
arXiv-issued DOI via DataCite
Submission history
From: Shafayeth Jamil [view email] [v1] Tue, 31 Mar 2026 04:59:56 UTC (1,012 KB) [v2] Wed, 1 Apr 2026 02:49:26 UTC (1,013 KB)
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