Lie Generator Networks for Nonlinear Partial Differential Equations

TL;DR

This paper introduces LGN-KM, a neural operator that linearizes nonlinear PDE dynamics in a spectral space, enabling interpretability, stability, and physics-informed modeling from trajectory data.

Contribution

The paper presents a novel neural operator architecture that learns a stable, interpretable Koopman generator for nonlinear PDEs, with applications to turbulence modeling.

Findings

Successfully recovers known dissipation scaling in turbulence

Extracts spectral structure without physics supervision

Enables long-horizon stability and model transfer

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.