Lie Generator Networks for Nonlinear Partial Differential Equations

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.

• The paper presents a novel Lie Generator Network-Koopman (LGN-KM) that lifts nonlinear PDEs into a latent linear space for direct spectral analysis.
• It enforces an S-D decomposition with a skew-symmetric matrix and positive-definite dissipation, guaranteeing modal interpretability and provable spectral stability.
• Experimental results demonstrate that LGN-KM recovers Navier-Stokes dissipation scaling with high R² values and supports efficient cross-regime transfer.

Lie Generator Networks for Koopman Linearization of Nonlinear PDE Dynamics

Introduction

The paper presents Lie Generator Network-Koopman (LGN-KM), a spectral neural operator that leverages a continuous-time Koopman generator with a Lie theoretic decomposition to describe and analyze the evolution of nonlinear PDEs in a linear latent space. This architecture enables direct access to the generator’s eigenspectrum, facilitating modal diagnostics and stability guarantees unprecedented in prior neural operator and Koopman learning frameworks.

Architecture and Methodological Contributions

LGN-KM employs a structured generator $L_k = S - D_k$, comprising a skew-symmetric, globally shared component $S$ and a wavenumber-dependent positive-definite diagonal $D_k$. The neural network pipeline integrates a nonlinear encoder implementing Koopman lifting, which projects the input temporal-spatial data into an $r$-dimensional latent channel space. The latent fields are Fourier transformed for spectral propagation.

• Generator construction: $S$ encodes conservative inter-modal (i.e., inter-channel) coupling, ensuring the presence of conjugate eigenvalue pairs (purely imaginary in the absence of dissipation). $D_k$ models mode-selective dissipation, parameterized to enforce viscous Laplacian scaling in the spectral domain.
• Propagation: The time evolution is computed by matrix exponentiation $C_t[k] = \exp(L_k t) C_0[k]$ at each wavenumber. This realizes a continuous-time solution operator and confines the computational cost to $\mathcal{O}(1)$ in the prediction horizon.
• Decoder and Training: A pointwise decoder projects the propagation back to observables, with the entire model trained end-to-end against trajectory reconstruction, without explicit physics-based loss or supervision.

This S-D decomposition both imposes and reveals interpretable physics constraints and guarantees long-horizon contractivity due to the negative real part of all generator eigenvalues.

Empirical Analysis on Nonlinear Dynamics

Recovery of Physical Laws from Data

On 2D Navier-Stokes turbulence, LGN-KM recovers the full modal dispersion relation and scaling laws intrinsic to the underlying PDEs. Notably:

• The dominant generator eigenvalue’s real part exhibits $|k|^2$ linear scaling, reconstructing the viscid dissipation term from data alone.
• Multiple dispersion branches, governed by $S$ (oscillation frequencies, bounded independently of $S$0) and $S$1 (decay rates, growing linearly in $S$2), are resolved and continuously parameterized by wavenumber.
• Every eigenvalue strictly satisfies $S$3, ensuring global stability.

Quantitatively, dissipation scaling fits achieve $S$4 across flow regimes. The consistent eigenspectral structures between independently trained models at different viscosities support the theoretical prediction of universality in the advective coupling (Kolmogorov’s similarity hypothesis).

Universality and Gauge Symmetry

The learned $S$5 matrices exhibit low raw cosine similarity across viscosities but nearly identical singular value spectra, signifying gauge freedom in the Koopman lifting: physical content is encoded in gauge-invariant eigenspectra rather than coordinate representations.

Similarly, cross-regime analysis confirms that the Laplacian dissipation coefficients $S$6 diverge as physical conditions change (reflecting viscosity dependence), while the base damping $S$7 and singular value profile of $S$8 remain nearly universal ($S$9 for $D_k$0-SVD and $D_k$1).

Generalization Beyond Fluid Dynamics

Applications to 2D FitzHugh-Nagumo reaction-diffusion show LGN-KM’s architectural priors generalize; the generator recovers the anticipated diffusive scaling of the dominant eigenvalue with $D_k$2 and universal conservative structure in $D_k$3, even with radically distinct nonlinearities and physical domains.

Stability, Continuous-Time Reasoning, and Model Transfer

The enforced generator structure enables unique utilities compared to conventional neural operators:

• Long-horizon stability: Unlike autoregressive baselines such as FNO, LGN-KM’s rollouts cannot diverge, as energy monotonically decays or is conserved in the latent linear system.
• Continuous-time prediction: Owing to the continuous parameter $D_k$4 in $D_k$5, predictions at arbitrary time horizons require only a single evaluation—bypassing the linear cost of sequential neural operator applications.
• Physics-informed transfer: The universal skew-symmetric part $D_k$6 can be frozen and transferred between regimes (e.g., low to high viscosity), dramatically reducing target data requirements and accelerating convergence in new physical regimes, which is grounded in the separation of universal (conservative) and regime-dependent (dissipative) generator components.

Comparison with Related Methods

LGN-KM extends and structurally differentiates from existing Koopman operator learning, neural operator, and structure-preserving modeling approaches:

• Deep Koopman and extended DMD methods learn (often unconstrained) discrete or continuous operators but typically forsake stability and direct spectral interpretability.
• SGNO constrains generator eigenvalues for stability but lacks oscillatory/conservative inter-channel structure due to diagonal-only dissipation.
• Hamiltonian, Lagrangian, and symplectic neural nets preserve invariants through implicit integration but do not directly parameterize or expose a spectral generator.

Critically, only the full S-D decomposition provides guaranteed stability, physically interpretable eigenspectra, and mechanism for universal structure transfer.

Limitations and Future Directions

• Accuracy trade-off: Imposing generator constraints reduces one-step forecast accuracy compared to unconstrained neural operators but yields interpretable and robust long-horizon behavior.
• Scalability: While the cost of the matrix exponential is negligible for moderate latent sizes ($D_k$7), scaling to very large $D_k$8 may require approximations or algorithmic improvements.
• Scope and extensibility: The current model leverages a per-mode Fourier decomposition, inherently tied to periodic boundary conditions in current experiments. Extensions to non-periodic, three-dimensional, or more heterogeneous domains remain open.
• Gauge ambiguity: The non-uniqueness of the Koopman lifting warrants future development of alignment or canonicalization procedures for direct parameter comparison and transfer.

Conclusion

LGN-KM establishes a new paradigm for neural operator learning of nonlinear PDEs, combining architectural guarantees of spectral accessibility, stability, and interpretable universal structure with practical computational advantages in long-horizon and cross-regime tasks. This approach promotes linear analysis of nonlinear dynamics, facilitating both theoretical investigation and robust simulation, and sets the stage for further development of physically structured, continuous-time machine learning representations for complex dynamical systems.

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For those pursuing theoretical and practical advances in operator learning, dynamical system identification, and physics-informed machine learning, the framework outlined in "Lie Generator Networks for Nonlinear Partial Differential Equations" (2603.29264) offers a robust, interpretable, and extensible foundation.