OP-00682 Solution claimed

Smooth-gradient approximation of finite-Fisher-information scores

Let $(M, g)$ be a smooth compact Riemannian manifold, possibly with smooth boundary, with volume measure $dV_g$. Let $\rho \ge 0$ satisfy $\int_M \rho \, dV_g = 1$ and $\sqrt\rho \in H^1(M)$, so that its Fisher information $$I(\rho) = 4 \int_M |\nabla \sqrt\rho|_g^2 \, dV_g$$ is finite, and define the score $s_\rho = 2 \nabla\sqrt\rho / \sqrt\rho$ on $\{\rho > 0\}$ and $s_\rho = 0$ elsewhere. One form of the exact chain rule for the entropy along curves of measures that Casteras, Flaim and Monsaingeon prove on such manifolds assumes that the score lies in the $L^2(\rho \, dV_g)$-closure of gradients of $C^1(M)$ functions; they showed that this holds when $\rho$ is bounded above and below by positive constants.

Problem. Do there exist $\varphi_j \in C^1(M)$, continuously differentiable up to the boundary, such that $$\int_M |\nabla \varphi_j - s_\rho|_g^2 \, \rho \, dV_g \longrightarrow 0 \qquad (j \to \infty)?$$ No convexity, uniform positivity, upper bound on $\rho$, or boundary condition on $\varphi_j$ is assumed.

Source: Casteras, Flaim and Monsaingeon (2025), the conjecture following Theorem 4.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Secondary: FA
Keywords: AIM Open Applied Problems, Fisher information, optimal transport, weighted Sobolev approximation

References

@misc{casteras2025,
  author        = {Casteras, Jean-Baptiste and Flaim, Marco and Monsaingeon, L{\'e}onard},
  title         = {Entropy and {F}isher information in non-convex domains: one chain to rule them all},
  year          = {2025},
  eprint        = {2512.05826},
  archivePrefix = {arXiv},
}
@misc{aim-fisher-score-gradient-closure,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Smooth-gradient approximation of finite-Fisher-information scores}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/665-fisher-score-gradient-closure.md},
  note         = {Entry 665 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Smooth-gradient approximation of Fisher scores by extension to a closed manifold v1 GitHub ↗

no authors listed

Yes: for every such (M, g) and rho there are phi_j smooth up to the boundary with the weighted L^2 error tending to zero, by Sobolev extension of sqrt(rho) to the double of M, Erbar's subdifferential theorem there, and restriction..

Full resolution · not verified · fidelity not established · posted on GitHub 1 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, 618-fisher-score/PROOF.md.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00682, "Smooth-gradient approximation of finite-Fisher-information scores", clopen.solutions.