Wasserstein gradient and Hamiltonian flows have emerged as essential tools for modeling complex dynamics in the natural sciences, with applications ranging from partial differential equations (PDEs) and optimal transport to quantum mechanics and information geometry. Despite their significance, the inverse identification of potential functions and interaction kernels underlying these flows remains relatively unexplored. In this work, we tackle this challenge by addressing the inverse problem of simultaneously recovering the potential function and interaction kernel from discretized observations of the density flow. We formulate the problem as an optimization task that minimizes a loss function specifically designed to enforce the underlying variational structure of Wasserstein flows, ensuring consistency with the geometric properties of the density manifold. Our framework employs a kernel-based operator approach using the associated Reproducing Kernel Hilbert Space (RKHS), which provides a closed-form representation of the unknown components. Furthermore, a comprehensive error analysis is conducted, providing convergence rates under adaptive regularization parameters as the temporal and spatial discretization mesh sizes tend to zero. Finally, a stability analysis is presented to bridge the gap between discrete trajectory data and continuous-time flow dynamics for the Wasserstein Hamiltonian flow.

翻译：Wasserstein梯度流与哈密顿流已成为自然科学中建模复杂动力学的重要工具，其应用范围涵盖偏微分方程（PDE）、最优传输、量子力学及信息几何等领域。尽管这些流具有重要价值，但对其背后势函数与相互作用核的逆向辨识研究仍相对不足。本研究通过从密度流的离散观测数据中同时恢复势函数与相互作用核的逆问题来应对这一挑战。我们将该问题构建为优化任务，通过最小化专门设计的损失函数来强化Wasserstein流的变分结构，确保与密度流形几何特性的一致性。该框架采用基于再生核希尔伯特空间（RKHS）的核算子方法，为未知分量提供闭式表示。此外，研究进行了完整的误差分析，在自适应正则化参数下给出了时空离散网格尺寸趋于零时的收敛速率。最后，针对Wasserstein哈密顿流提出了稳定性分析，以弥合离散轨迹数据与连续时间流动力学之间的理论鸿沟。