Neural and spectral operators can approximate shape-to-solution maps for families of elliptic and parabolic PDEs and BIEs with provable uniform error bounds derived from parametric holomorphy on a reference domain.
Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations
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abstract
We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.
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2026 2verdicts
UNVERDICTED 2representative citing papers
The Root Theorem states that context engineering is governed by maximizing signal-to-token ratio inside finite, lossy channels, which forces homeostatic architectures of accumulate-compress-rewrite-shed and external verification.
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Neural Shape Operator Surrogates -- Expression Rate Bounds
Neural and spectral operators can approximate shape-to-solution maps for families of elliptic and parabolic PDEs and BIEs with provable uniform error bounds derived from parametric holomorphy on a reference domain.
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The Root Theorem of Context Engineering
The Root Theorem states that context engineering is governed by maximizing signal-to-token ratio inside finite, lossy channels, which forces homeostatic architectures of accumulate-compress-rewrite-shed and external verification.