Convergence rate of solutions under cylindrical approximation

Determine the quantitative rate at which the cylindrical-approximated solution F([P_m θ], t) converges to the exact solution F([θ], t) as m increases, for abstract evolution equations on Banach spaces of functionals generated by closed, densely-defined, continuous linear operators L([]), under the assumptions of stability and consistency of the approximation.

Background

The paper establishes conditions under which the cylindrical approximation of abstract evolution equations is consistent and stable and proves convergence of the approximated solution to the exact one as the degree m increases. While convergence (qualitative) is shown, the authors note that a quantitative characterization of how fast this convergence occurs has not been provided.

A convergence rate for the solution is essential for error control, algorithmic design, and resource allocation, especially when using physics-informed neural networks to solve the resulting high-dimensional PDEs. The existing results provide rates for functional and Fréchet derivatives but not for the solution itself.

References

To our knowledge, the convergence rate has been unknown so far.

— Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees  (2410.18153 - Miyagawa et al., 2024) in Appendix: Theoretical Background of Cylindrical Approximation and Convergence, Abstract Evolution Equations

We have not established a quantitative estimate, analogous to the O(ε{2}) convergence rate proved by Arrighi, Nesme, and Forets for the higher-dimensional Archimedean DTQW--Dirac correspondence , for the convergence of the discretized CTQW on G_{l}{3} (Section \ref{Section_CTQW_II}) to the full p-adic Dirac equation as l\rightarrow\infty. Since \mathbb{Z}_{p}{3} is compact and ultrametric, one may expect a rate governed by p{-l} rather than by a continuous ε; making this precise is open.

— p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks  (2609.04358 - Zúñiga-Galindo, 3 Sep 2026) in Section 6, subsection “Open problems,” item (1) (Quantitative continuum limit)