AI 中文总结
DisKO通过联合学习分布可观测函数、有限维库普曼表示和生成映射，在分布空间中实现深度库普曼学习，在七个基准上达到最优外推性能并显著降低长时程误差累积。
AI 中文摘要
许多复杂系统仅通过时间上非配对的分布快照被观测到，这使得在没有额外假设的情况下进行基于轨迹的动力学学习变得困难。因此，我们直接在分布空间中表述该问题，将分布本身视为动力学状态。挑战在于分布空间是无限维的，使得从有限快照中学习紧凑且近似封闭的表示变得困难。我们引入了DisKO，它将深度库普曼学习扩展到分布动力学，通过联合学习预测性分布可观测函数、一个有限维库普曼表示，以及一个映射回完整分布的生成映射。在七个不同的基准测试中，DisKO实现了最先进的外推性能，在长时程预测任务中误差累积显著更慢。DisKO进一步在具有解析谱的系统上恢复了主要的库普曼特征值和特征函数，揭示了学习表示中有意义的动力学结构。
英文摘要
Many complex systems are observed only through temporally unpaired distribution snapshots, making trajectory-based dynamical learning difficult without additional assumptions. We therefore formulate the problem directly in distribution space, treating the distribution itself as the dynamical state. The challenge is that distribution space is infinite-dimensional, making compact and approximately closed representations difficult to learn from finite snapshots. We introduce DisKO, which extends deep Koopman learning to distribution dynamics by jointly learning predictive distributional observables, a finite-dimensional Koopman representation, and a generative map back to the full distribution. Across seven diverse benchmarks, DisKO achieves state-of-the-art extrapolation performance, with substantially slower error accumulation on long-horizon prediction tasks. DisKO further recovers leading Koopman eigenvalues and eigenfunctions on systems with analytic spectra, revealing meaningful dynamical structure in the learned representation.
