Lie Generator Networks for Nonlinear Partial Differential Equations
This work addresses the lack of a theoretical framework for nonlinear PDEs, offering a method for interpretable and stable modeling in fluid dynamics, though it is incremental as it builds on Koopman operator theory.
The authors tackled the problem of characterizing nonlinear partial differential equations by introducing Lie Generator Network-Koopman (LGN-K), which lifts nonlinear dynamics into a linear latent space and learns a stable, interpretable generator; on two-dimensional Navier-Stokes turbulence, it recovered known dissipation scaling and a multi-branch dispersion relation from data alone.
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.