A kernel method for the learning of Wasserstein geometric flows
Jianyu Hu, Juan-Pablo Ortega, Daiying Yin
TL;DR
This work tackles the inverse problem of learning the energy landscape that drives Wasserstein gradient and Hamiltonian flows on the density manifold from discretized density trajectories. It introduces a structure-preserving kernel ridge regression framework in RKHS spaces to simultaneously recover the potential $V$, the interaction kernel $W$, and, when available, the internal energy $U$, yielding closed-form estimators via a differential reproducing property. The analysis provides explicit error decompositions into approximation, mesh-induced, and scheme-dependent terms, and establishes convergence rates under adaptive regularization and discretization, together with stability results showing uniform convergence of the learned Hamiltonian flow to the true continuous flow in Wasserstein distance. By avoiding basis truncation and exploiting the intrinsic geometry of Wasserstein manifolds, the approach yields a theoretically solid and computationally tractable method for data-driven discovery of energy functionals in infinite-dimensional settings with applications to PDEs, kinetic models, and mean-field dynamics.
Abstract
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.