Proves the first universal approximation theorems for k-times differentiable nonlinear operators between Banach spaces and their derivatives uniformly on compact sets in weighted Sobolev norms via encoder-decoder operator learning architectures.
Shape derivative- informed neural operators with application to risk-averse shape optimization
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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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Universal Approximation of Nonlinear Operators and Their Derivatives
Proves the first universal approximation theorems for k-times differentiable nonlinear operators between Banach spaces and their derivatives uniformly on compact sets in weighted Sobolev norms via encoder-decoder operator learning architectures.
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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.