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.
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs.arXiv preprint arXiv:2004.01806, 2020
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Establishes convergence for non-Lipschitz generators via bounded double-well lemma and truncated BSDE analysis, plus XNet architecture for efficient 100D PDE computation.
A systematic review of Kolmogorov-Arnold Networks that maps their relation to Kolmogorov superposition theory, MLPs, and kernels, examines basis-function design choices, summarizes performance advances, and supplies a practitioner's selection guide plus open challenges.
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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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XNet-Enhanced Deep BSDE Method and Numerical Analysis
Establishes convergence for non-Lipschitz generators via bounded double-well lemma and truncated BSDE analysis, plus XNet architecture for efficient 100D PDE computation.
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A Practitioner's Guide to Kolmogorov-Arnold Networks
A systematic review of Kolmogorov-Arnold Networks that maps their relation to Kolmogorov superposition theory, MLPs, and kernels, examines basis-function design choices, summarizes performance advances, and supplies a practitioner's selection guide plus open challenges.