Lie Generator Networks for Nonlinear Partial Differential Equations

2026-03-31

AI Summary

Existing approaches struggle to establish a comprehensive characterization of nonlinear partial differential equation systems analogous to linear spectral theory. This work proposes the LGN-KM architecture, which embeds nonlinear dynamics into a linear latent space via neural operators and learns a continuous-time Koopman generator in an unsupervised manner. By imposing Lie algebraic constraints—specifically skew-symmetry and positive-definite diagonal decomposition—the generator is decoupled into conservative coupling and modal dissipation components. The method uniquely recovers turbulent dissipation laws and multi-branch dispersion relations directly from trajectory data without requiring physical priors, thereby revealing the gauge freedom inherent in Koopman embeddings. Applied to two-dimensional Navier–Stokes turbulence, the model accurately reproduces known physical laws, enables cross-viscosity generalization, supports continuous prediction at arbitrary times, and maintains long-term numerical stability.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• nonlinear partial differential equations
• Koopman generator
• dynamical systems
• spectral analysis
• stability

Innovation

• Lie Generator Network
• Koopman operator
• neural operator
• spectral decomposition
• nonlinear PDEs