Learning the Energy Landscapes of Dynamical Systems via Energetic Variational Optimal Transport under Data Quantity--Quality Trade-offs

Dynamic optimal transport unifies optimal transport, fluid mechanics, and gradient-flow theory within a continuous dynamical framework, offering a geometry-aware language for applications across physics, biology, and machine learning.

However, conventional formulations cast it as a constrained optimization problem that must explicitly satisfy the continuity equation, hindering the reconstruction of the underlying dynamics directly from data.

We propose the energetic variational method for dynamic optimal transport (EVMDOT), which reformulates it within an energetic variational framework by combining the flow map, the least action principle, and the maximum dissipation principle.

The flow map recasts the constrained problem as an unconstrained one by automatically enforcing the continuity equation, while the balance between the conservative and dissipative forces determines the velocity field.

Applied to the Fokker--Planck equation, the EVMDOT reconstructs both the energy landscape and the Waddington landscape directly from time-series density data.

Through numerical experiments, we reveal that the EVMDOT achieves an intrinsic balance between data quantity and data quality: a sufficient data quantity compensates for limited data quality, making the reconstruction robust to the choice of the observation window.

We further apply the EVMDOT to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset to infer the potential landscape of amyloid-$\beta$ and tau, revealing two wells corresponding to the cognitively normal and Alzheimer's disease stages and the transition pathway between them.