# Universal Approximation of Nonlinear Operators and Their Derivatives

arXiv: https://arxiv.org/abs/2605.15285 · HTML: https://mlsanity.com/paper/2605.15285/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 47 of its 185 references that are in the map.
- First posted on arXiv 14 May 2026.

## Builds on (most cited first, 30 of 47)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Learning in Mean Field Games: A Survey](https://mlsanity.com/paper/2205.12944/) (2022)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs](https://mlsanity.com/paper/2011.15110/) (2020)
- [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators](https://mlsanity.com/paper/2305.20053/) (2023)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Derivative-enhanced Deep Operator Network](https://mlsanity.com/paper/2402.19242/) (2024)
- [Mean-field neural networks: learning mappings on Wasserstein space](https://mlsanity.com/paper/2210.15179/) (2022)
- [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024)
- [Minimal Width for Universal Property of Deep RNN](https://mlsanity.com/paper/2211.13866/) (2022)
- [Accurate, scalable, and efficient Bayesian optimal experimental design with derivative-informed neural operators](https://mlsanity.com/paper/2312.14810/) (2023)
- [Sequential infinite-dimensional Bayesian optimal experimental design with derivative-informed latent attention neural operator](https://mlsanity.com/paper/2409.09141/) (2024)
- [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024)
- [Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimization](https://mlsanity.com/paper/2512.14086/) (2025)
- [Performance of Neural and Polynomial Operator Surrogates](https://mlsanity.com/paper/2604.00689/) (2026)
- [A PINN approach for the online identification and control of unknown PDEs](https://mlsanity.com/paper/2408.03456/) (2024)
- [Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees](https://mlsanity.com/paper/2410.18153/) (2024)
- [Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization](https://mlsanity.com/paper/2603.03211/) (2026)

## 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).