Universal Approximation of Nonlinear Operators and Their Derivatives
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Abstract (arXiv)
We show that Universal Approximation (UA) of nonlinear operators and their derivatives via Operator Learning (OL) architectures fails in CkF (Fr'echet) compact-open topologies and in Fr'echet--Sobolev norms (i.e. under operator norms). We solve this obstruction by restoring UA in natural weaker topologies: CkB (Bastiani) compact-open topologies and (novel) weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional spaces and OL. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. These results launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives) on general Banach spaces. We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures available in general Banach spaces; these include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. A key mathematical result is that our new weighted Bastiani--Sobolev spaces generalize classical Gaussian (Malliavin) Sobolev spaces on Banach spaces. Open frontiers where DIOL and our UATs find applications are: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control).
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References · 47
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
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- 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
- 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
- 2019Universal Approximation with Deep Narrow Networkscited by 200
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- 2017Sobolev Training for Neural Networkscited by 143
- 2024Operator Learning: Algorithms and Analysiscited by 85
- 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
- 2022Learning in Mean Field Games: A Surveycited by 61
- 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 44
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- 2024LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transportcited by 7
- 2025Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimizationcited by 7
- 2026Performance of Neural and Polynomial Operator Surrogatescited by 7
- 2024A PINN approach for the online identification and control of unknown PDEscited by 5
- 2024Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guaranteescited by 5
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- 2026The Volterra signaturecited by 5
- 2025New universal operator approximation theorem for encoder-decoder architecturescited by 4
- 2025Dimension reduction for derivative-informed operator learning: An analysis of approximation errorscited by 4
- 2026Universal approximation with signatures of non-geometric rough pathscited by 4
- 2023Optimal adaptive control with separable drift uncertaintycited by 3
- 2024Neural networks in non-metric spacescited by 3
- 2026Learning operators on labelled conditional distributions with applications to mean field control of non exchangeable systemscited by 3
- 2026PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methodscited by 3
- 2023Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic modelscited by 2
- 2024Finite Dimensional Projections of HJB Equations in the Wasserstein Spacecited by 2
- 2024Optimal control of partial differential equations in PyTorch using automatic differentiation and neural network surrogatescited by 2
- 2025Optimal control of Volterra integral diffusions and application to contract theorycited by 2
- 2025Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applicationscited by 2
- 2026Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operatorscited by 2
- 2026Topological DeepONets and a generalization of the Chen-Chen operator approximation theoremcited by 2
- 2026Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Controlcited by 2
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