We introduce a machine learning framework for modeling the spatio-temporal dynamics of mass-constrained complex systems with hidden states, whose behavior can, in principle, be described by PDEs but lack explicit models. The method extends the Equation-Free approach, enabling the data-driven reconstruction of reduced-order models (ROMs) without needing to identify governing equations. Using manifold learning, we obtain a latent space representation of system evolution from data via delayed coordinates, in accordance with Takens/Whitney's embedding theorems. Linear (Proper Orthogonal Decomposition, POD) and nonlinear (Diffusion Maps, DMs) methods are employed to extract low-dimensional embeddings that capture the essential dynamics. Predictive ROMs are then learned within this latent space, and their evolution is lifted back to the original high-dimensional space by solving a pre-image problem. We show that both POD and k-nearest neighbor (k-NN) lifting operators preserve mass, a key physical constraint in systems such as computational fluid dynamics and crowd dynamics. Our framework effectively reconstructs the solution operator of the underlying PDE without discovering the PDE itself, by leveraging a manifold-informed objective map that bridges multiple scales. For our illustrations, we use synthetic spatio-temporal data from the Hughes model, which couples a continuity PDE with an Eikonal equation describing optimal path selection in crowds. Results show that DM-based nonlinear embeddings outperform POD in reconstruction accuracy, producing more parsimonious and stable ROMs that remain accurate and integrable over long time horizons.

翻译：我们提出了一种机器学习框架，用于建模具有隐状态且满足质量守恒约束的复杂系统时空动力学。此类系统的行为原则上可用偏微分方程描述，但缺乏显式模型。该方法扩展了"无方程"框架，无需识别控制方程即可实现数据驱动的降阶模型重构。基于流形学习理论，我们通过延迟坐标从数据中获取系统演化的隐空间表示，这符合Takens/Whitney嵌入定理。采用线性方法（本征正交分解）和非线性方法（扩散映射）提取捕获系统本质动力学的低维嵌入。在此隐空间中学习预测性降阶模型，并通过求解原像问题将其演化提升回原始高维空间。我们证明POD和k近邻提升算子均能保持质量守恒——这是计算流体力学和人群动力学等系统中的关键物理约束。该框架通过构建连接多尺度的流形信息目标映射，有效重构了底层偏微分方程的解算子而无需发现方程本身。在示例中，我们使用Hughes模型生成的合成时空数据，该模型将连续性偏微分方程与描述人群最优路径选择的Eikonal方程耦合。结果表明：基于扩散映射的非线性嵌入在重构精度上优于POD方法，能产生更简约稳定的降阶模型，并在长时间尺度上保持准确性和可积分性。