# Variable-Input Deep Operator Networks

arXiv: https://arxiv.org/abs/2205.11404 · HTML: https://mlsanity.com/paper/2205.11404/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 23 May 2022.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Set Transformer: A Framework for Attention-based Permutation-Invariant Neural Networks](https://mlsanity.com/paper/1810.00825/) (2018)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)

## Built on by (most cited first, 25 of 25)

- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains](https://mlsanity.com/paper/2305.19663/) (2023)
- [Variationally Mimetic Operator Networks](https://mlsanity.com/paper/2209.12871/) (2022)
- [Space-Time Continuous PDE Forecasting using Equivariant Neural Fields](https://mlsanity.com/paper/2406.06660/) (2024)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)
- [ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines](https://mlsanity.com/paper/2501.09395/) (2025)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [StFT: Spatio-temporal Fourier Transformer for Long-term Dynamics Prediction](https://mlsanity.com/paper/2503.11899/) (2025)
- [SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling](https://mlsanity.com/paper/2505.04738/) (2025)
- [Deep set based operator learning with uncertainty quantification](https://mlsanity.com/paper/2509.25646/) (2025)
- [BLISSNet: Deep Operator Learning for Fast and Accurate Flow Reconstruction from Sparse Sensor Measurements](https://mlsanity.com/paper/2602.24228/) (2026)
- [DETNO: A Diffusion-Enhanced Transformer Neural Operator for Long-Term Traffic Forecasting](https://mlsanity.com/paper/2508.19389/) (2025)
- [In-Context Multi-Operator Learning with DeepOSets](https://mlsanity.com/paper/2512.16074/) (2025)
- [Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces](https://mlsanity.com/paper/2608.06428/) (2026)
- [Deep operator learning for efficient sampling from invariant measures of stochastic differential equations](https://mlsanity.com/paper/2609.11376/) (2026)

## Abstract (arXiv)

Existing architectures for operator learning require that the number and locations of sensors (where the input functions are evaluated) remain the same across all training and test samples, significantly restricting the range of their applicability. We address this issue by proposing a novel operator learning framework, termed Variable-Input Deep Operator Network (VIDON), which allows for random sensors whose number and locations can vary across samples. VIDON is invariant to permutations of sensor locations and is proved to be universal in approximating a class of continuous operators. We also prove that VIDON can efficiently approximate operators arising in PDEs. Numerical experiments with a diverse set of PDEs are presented to illustrate the robust performance of VIDON in learning operators.