Lie Generator Networks for Nonlinear Partial Differential Equations

Most physical systems are governed by nonlinear partial differential equations, yet the analytical tools that make linear systems transparent, such as eigenspectra, dispersion relations and modal decomposition, remain unavailable in nonlinear settings. 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 through the decomposition $L_k = S - D_k$, where $S$ is skew-symmetric (conservative coupling) and $D_k$ is positive-definite diagonal (mode-selective dissipation). The decomposition guarantees stability by construction and exposes the eigenstructure directly, making linear-systems analysis available on a nonlinear PDE. On 2D Navier-Stokes turbulence, the generator recovers a complete multi-branch dispersion relation and the viscous dissipation scaling $\mathrm{Re}(\lambda) \propto -|k|^2$ from trajectory data alone, with no physics supervision. The same architecture recovers the analogous diffusive scaling on FitzHugh-Nagumo reaction-diffusion without modification. Independently trained models at different regimes show matched gauge-invariant spectral structure. The architecture additionally enables long-horizon stability, $O(1)$ continuous-time evaluation, and physics--informed cross-regime model transfer.