Universal Approximation of Nonlinear Operators and Their Derivatives
The paper titled 'Universal Approximation of Nonlinear Operators and Their Derivatives' explores the concept of Derivative-Informed Operator Learning (DIOL). It presents the first Universal Approximation Theorems (UATs) for nonlinear operators and their derivatives in infinite-dimensional settings. The findings have implications for various applications, including optimal control of partial differential equations and numerical methods.
- ▪The paper proves the first UATs of non-linear k-times differentiable operators between Banach spaces and their derivatives.
- ▪These results generalize classical results to infinite-dimensional settings and operator learning architectures.
- ▪The research discusses applications in high-order accuracy, fast constrained optimization, and numerical methods for infinite-dimensional PDEs.
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Computer Science > Machine Learning arXiv:2605.15285 (cs) [Submitted on 14 May 2026] Title:Universal Approximation of Nonlinear Operators and Their Derivatives Authors:Filippo de Feo View a PDF of the paper titled Universal Approximation of Nonlinear Operators and Their Derivatives, by Filippo de Feo View PDF HTML (experimental) Abstract:Derivative-Informed Operator Learning (DIOL), i.e. learning a (nonlinear) operator and its derivatives, is an open research frontier at the foundations of the influential field of Operator Learning (OL). In particular, Universal Approximation Theorems (UATs) of nonlinear operators and their derivatives are foundational open questions and delicate problems in nonlinear functional analysis.
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