用神经网络学习非线性方程的线性生成器，实现稳定且可解释的长期预测。
Lie Generator Networks for Nonlinear Partial Differential Equations
- 将非线性动力系统映射到线性潜空间，分解生成器为保守耦合与耗散项。
- 仅用轨迹数据恢复纳维-斯托克斯湍流的耗散规律和多分支色散关系。
- 适合需要长期稳定预测与物理可解释性的流体建模研究者。
线性动力系统由其特征谱完全刻画，可通过动力学生成器直接获得。对于由偏微分方程支配的非线性系统，尚无等价理论。本文提出李生成器网络-库普曼（LGN-KM），一种神经算子，将非线性动力系统提升至线性潜空间，并通过分解 $L_k = S - D_k$ 学习连续时间库普曼生成器：$S$ 为斜对称部分，代表保守的模态间耦合；$D_k$ 为正定对角部分，编码模态耗散。该结构强制稳定性并实现谱级可解释性。在二维纳维-斯托克斯湍流上，仅用轨迹数据即恢复已知耗散标度律和完整的多分支色散关系，无需物理监督。不同流动状态独立训练的模型均恢复一致的规范不变谱结构，揭示了库普曼提升中的规范自由度。由于生成器可证明稳定，支持保证的长时序稳定性、任意时刻的连续时间评估及物理信息引导的跨粘度模型迁移。
原文摘要 · Abstract (English)
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.
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