AI 中文总结
本研究针对混沌系统提出概率非侵入式降阶模型，用非线性自编码器结合隐空间SDE建模，生成的轨迹保留统计特性，为混沌研究提供可靠替代方案
AI 中文摘要
随机微分方程（SDE）已成为科学机器学习的基石，不过它们主要被用作不确定性量化或分布匹配的算法工具。相比之下，将SDE从根本上作为随机过程来建模宏观非线性物理的研究仍大多未被探索。本研究针对混沌动力系统提出了一种概率型非侵入式降阶模型（ROM）。我们认为，将高维非线性动力学投影到低维流形上会引入不可约不确定性，这种不确定性因湍流流动固有的混沌吸引子和多可容许未来状态而加剧。因此，由偏微分方程控制的混沌系统可在合适的隐空间中通过SDE进行有效建模。为此，我们采用非线性自编码器将流场映射为低维表示，其时间演化由SDE显式控制。动力学的可预测部分由学习到的漂移项捕获，而状态依赖的随机性则由扩散项吸收。我们证明，该概率框架可成功传播高度非线性状态，为混沌 regime 提供了传统确定性方法的可靠替代方案。最终，我们的模型生成的新混沌流轨迹与从直接数值模拟（DNS）数据中学习到的真实转移核在局部和全局上保持一致。尽管这些生成的轨迹是唯一的且不同于训练集，但它们保留了潜在的统计特性和流形，验证了我们方法的强大生成性能和鲁棒性。
英文摘要
Stochastic Differential Equations (SDEs) have become a cornerstone of scientific machine learning, though they are predominantly utilized as algorithmic tools for uncertainty quantification or distribution matching. In contrast, leveraging SDEs fundamentally to model macroscopic, nonlinear physics as stochastic processes remains largely unexplored. This work introduces a probabilistic, non-intrusive reduced-order model (ROM) for chaotic dynamical systems. We argue that projecting high-dimensional nonlinear dynamics onto a low-dimensional manifold introduces irreducible uncertainty, compounded by the chaotic attractors and multi-admissible futures inherent to turbulent flows. Consequently, a chaotic system governed by a partial differential equation can be effectively modeled by an SDE in a suitable latent space. To this end, a nonlinear autoencoder is employed to map the flow field into a low-dimensional representation, within which the temporal evolution is explicitly governed by an SDE. The predictable component of the dynamics is captured by a learned drift term, while state-dependent stochasticity is absorbed by a diffusion term. We demonstrate that this probabilistic framework successfully propagates highly nonlinear states, offering a robust alternative to traditional deterministic methodologies for chaotic regimes. Ultimately, our model generates new chaotic flow trajectories that remain locally and globally consistent with the true transition kernel learned from Direct Numerical Simulation (DNS) data. Even though these generated trajectories are unique and distinct from the training set, they preserve the underlying statistics and manifolds, validating the strong generative performance and robustness of our methodology.
