PROMETHEUS: A Coupled Wasserstein--Riemannian Variational Model: Mathematical Setup, Dynamics, and Energy Dissipation
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2026
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| author | Ben Ammar Chraiti, Mohamed |
| author_facet | Ben Ammar Chraiti, Mohamed |
| contents | <div>PROMETHEUS is a mathematical research manuscript introducing a coupled Wasserstein--Riemannian variational model in which a probability measure $\mu$ and a Riemannian metric $g$ evolve simultaneously on a smooth closed manifold $\mathcal{M}$ under a common energy functional.</div> <div> </div> <div>The total energy is</div> <div>$\mathcal{F}[\mu,g] = \mathcal{E}[\mu,g] + \lambda \mathcal{R}[g] + \kappa \mathcal{S}[g]$,</div> <div>where $\mathcal{E}$ is a nonlocal kernel-weighted entropy, $\mathcal{R}[g]$ is the Einstein--Hilbert functional, and $\mathcal{S}[g]$ is a coercive Sobolev-type penalty relative to a reference metric $g_{\mathrm{ref}}$.</div> <div> </div> <div>The manuscript develops the full variational structure of the model. In particular, it computes the first variation of $\mathcal{F}$ with respect to the measure variable $\mu$ and with respect to the metric variable $g$, taking into account the fact that the density</div> <div>$\rho_g = d\mu / d\mathrm{vol}_g$</div> <div>depends implicitly on the metric through the Riemannian volume form. The entropy term is built from an admissible smoothing kernel $K_{\varepsilon,g}$, with canonical examples given by the heat kernel and a localized distance-based kernel.</div> <div> </div> <div>The associated dynamics are formulated as a coupled gradient-flow system. The measure component is a Wasserstein-type continuity equation, while the metric component is a gradient descent equation on the space of Riemannian metrics. A DeTurck gauge-fixing term is introduced in the metric equation in order to remove the diffeomorphism degeneracy of the geometric part and restore strict parabolicity at the level of the principal symbol.</div> <div> </div> <div>A central part of the manuscript is the derivation of a formal energy balance identity. The DeTurck modification produces a correction term $\mathcal{C}_{\mathrm{DeT}}(t)$ in the energy balance. The paper proves two dissipation regimes: exact energy dissipation for the fully modified diffeomorphism-covariant system when $\kappa = 0$, and approximate dissipation for the partially modified system under explicit a priori regularity assumptions.</div> <div> </div> <div>The work also studies stationary solutions, long-time behavior, and the structure of the critical set. It introduces a Lie--Trotter splitting scheme that alternates a JKO-type step in measure space with a metric gradient step, and it establishes formal short-time existence together with conditional uniqueness under additional regularity and Lipschitz assumptions.</div> <div> </div> <div>In addition, the manuscript includes a numerical implementation on the flat two-dimensional torus $T^2$, using a full-tensor discretization of the metric, a Sinkhorn-regularized approximation of the JKO step, a DeTurck-corrected metric update, and a strict admissibility projection. The numerical section is presented as a structurally faithful computational realization of the theoretical framework rather than as a complete convergence theory.</div> <div> </div> <div>Overall, the manuscript proposes a unified variational framework linking optimal transport, geometric evolution equations, and adaptive metric dynamics within a single coupled dissipative model.</div> <div> </div> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19890524 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | PROMETHEUS: A Coupled Wasserstein--Riemannian Variational Model: Mathematical Setup, Dynamics, and Energy Dissipation Ben Ammar Chraiti, Mohamed <div>PROMETHEUS is a mathematical research manuscript introducing a coupled Wasserstein--Riemannian variational model in which a probability measure $\mu$ and a Riemannian metric $g$ evolve simultaneously on a smooth closed manifold $\mathcal{M}$ under a common energy functional.</div> <div> </div> <div>The total energy is</div> <div>$\mathcal{F}[\mu,g] = \mathcal{E}[\mu,g] + \lambda \mathcal{R}[g] + \kappa \mathcal{S}[g]$,</div> <div>where $\mathcal{E}$ is a nonlocal kernel-weighted entropy, $\mathcal{R}[g]$ is the Einstein--Hilbert functional, and $\mathcal{S}[g]$ is a coercive Sobolev-type penalty relative to a reference metric $g_{\mathrm{ref}}$.</div> <div> </div> <div>The manuscript develops the full variational structure of the model. In particular, it computes the first variation of $\mathcal{F}$ with respect to the measure variable $\mu$ and with respect to the metric variable $g$, taking into account the fact that the density</div> <div>$\rho_g = d\mu / d\mathrm{vol}_g$</div> <div>depends implicitly on the metric through the Riemannian volume form. The entropy term is built from an admissible smoothing kernel $K_{\varepsilon,g}$, with canonical examples given by the heat kernel and a localized distance-based kernel.</div> <div> </div> <div>The associated dynamics are formulated as a coupled gradient-flow system. The measure component is a Wasserstein-type continuity equation, while the metric component is a gradient descent equation on the space of Riemannian metrics. A DeTurck gauge-fixing term is introduced in the metric equation in order to remove the diffeomorphism degeneracy of the geometric part and restore strict parabolicity at the level of the principal symbol.</div> <div> </div> <div>A central part of the manuscript is the derivation of a formal energy balance identity. The DeTurck modification produces a correction term $\mathcal{C}_{\mathrm{DeT}}(t)$ in the energy balance. The paper proves two dissipation regimes: exact energy dissipation for the fully modified diffeomorphism-covariant system when $\kappa = 0$, and approximate dissipation for the partially modified system under explicit a priori regularity assumptions.</div> <div> </div> <div>The work also studies stationary solutions, long-time behavior, and the structure of the critical set. It introduces a Lie--Trotter splitting scheme that alternates a JKO-type step in measure space with a metric gradient step, and it establishes formal short-time existence together with conditional uniqueness under additional regularity and Lipschitz assumptions.</div> <div> </div> <div>In addition, the manuscript includes a numerical implementation on the flat two-dimensional torus $T^2$, using a full-tensor discretization of the metric, a Sinkhorn-regularized approximation of the JKO step, a DeTurck-corrected metric update, and a strict admissibility projection. The numerical section is presented as a structurally faithful computational realization of the theoretical framework rather than as a complete convergence theory.</div> <div> </div> <div>Overall, the manuscript proposes a unified variational framework linking optimal transport, geometric evolution equations, and adaptive metric dynamics within a single coupled dissipative model.</div> <div> </div> <p> </p> |
| title | PROMETHEUS: A Coupled Wasserstein--Riemannian Variational Model: Mathematical Setup, Dynamics, and Energy Dissipation |
| url | https://doi.org/10.5281/zenodo.19890524 |