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Description
This work introduces a novel dynamical reduced-order approximation framework for Wasserstein gradient flows that leverages the geometric structure of the solution manifold to construct an adaptive low-dimensional representation. The proposed method evolves the solution parametrization through appropriately designed systems of ordinary differential equations, allowing the approximation space itself to evolve in time, adapting to the underlying solution manifold. Such evolution is optimal with respect to the metric induced on the tangent space, ensuring an efficient and accurate representation of the system dynamics.
Wasserstein gradient flows arise in a wide range of applications spanning advection-diffusion PDEs and optimization, and their numerical treatment becomes particularly challenging in high-dimensional settings. In this regime, their approximation is challenged by the curse of dimensionality, the prohibitive cost of optimal transport computations, and sampling inefficiencies in the representation of evolving probability measures. These difficulties are compounded by nonlinear interactions and the simultaneous presence of diffusive and transport-dominated dynamics. In contrast, the proposed dynamical approach is mesh-free, achieves high accuracy with a small number of basis, and requires only a single sampling of the initial condition.
As a result, we provide a general nonlinear reduced-order model capable of accurately approximating a broad class of physical time-dependent phenomena, demonstraing ts effectiveness on several prototypical Wasserstein gradient flows.