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Lie Generator Networks for Nonlinear Partial Differential Equations

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural ODE training and analysis · 0 citations (OpenAlex)

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Proposes LGN-KM, a neural operator that learns a continuous-time Koopman generator decomposed as skew-symmetric minus positive diagonal (S−D) per Fourier mode, yielding stable, interpretable spectra for 2D Navier–Stokes and enabling long-horizon stability and cross-viscosity transfer.

Problem

Nonlinear PDEs lack analytical tools such as eigenspectra and dispersion relations; existing neural operators predict solutions without exposing an interpretable, stable generator of the dynamics.

Approach

Learn the continuous-time Koopman generator per Fourier mode as L_k = S − D_k, with S skew-symmetric and shared across modes (conservative coupling) and D_k positive diagonal with wavenumber-dependent dissipation, so eigenvalues have non-positive real parts by construction and propagation is a single matrix exponential.

Key results

• Learned generator's dominant eigenvalue scales linearly with |k|^2 (R^2=0.920 at nu=1e-5, 0.947 at nu=1e-3), recovering viscous dissipation scaling without physics supervision.
• Freezing S from a nu=1e-5 model and fine-tuning the rest gives ~6x convergence speedup and 18% lower final error at N=50 on nu=1e-3 data; transfer at N=50 matches scratch at N=100.
• FNO is more accurate within the training horizon (relative L2 0.16 vs 0.24 at nu=1e-5), while LGN-KM energy stays bounded (~0.28) over 200 steps where FNO energy diverges to ~1e18.

Surprising findings

• Independently trained models at different viscosities learn different raw S matrices (cosine similarity 0.023) yet match gauge-invariant singular value profiles (R^2=0.941).
• Unconstrained generator yields unstable eigenvalues at every epoch (max Re up to +0.35), while the S-D version has similar test error (0.240 vs 0.247).

Limitations

• Less accurate than FNO within the training horizon (relative L2 0.24 vs 0.16 at nu=1e-5).
• Restricted to 2D periodic domains via Fourier-mode factorization; not demonstrated in 3D or non-periodic settings.
• Matrix exponential cost O(r^3) per mode limits scaling to large latent dimensions.

Experimental setup

Datasets: 2D Navier-Stokes benchmark (Li et al.) 2D FitzHugh-Nagumo simulated trajectories

Benchmarks: 2D Navier-Stokes turbulence (nu=1e-5, nu=1e-3) 2D FitzHugh-Nagumo

Metrics: relative L2 error eigenvalue spectra R^2 of dissipation scaling R^2 of singular value profiles wall time enstrophy over long horizon

Baselines: FNO-2D SGNO unconstrained generator S-only ablation D-only ablation scratch training

Builds on: Fourier Neural Operator Koopman operator / deep Koopman LGN for linear system identification CT-KAE

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