# DeepONet Augmented by Randomized Neural Networks for Efficient Operator Learning in PDEs

arXiv: https://arxiv.org/abs/2503.00317 · HTML: https://mlsanity.com/paper/2503.00317/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 43 references that are in the map.
- First posted on arXiv 1 Mar 2025.

## Builds on (most cited first, 16 of 16)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [LSTM Fully Convolutional Networks for Time Series Classification](https://mlsanity.com/paper/1709.05206/) (2017)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solving](https://mlsanity.com/paper/2408.02198/) (2024)
- [TransNet: Transferable Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2301.11701/) (2023)
- [Fast Function to Function Regression](https://mlsanity.com/paper/1410.7414/) (2014)
- [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)

## Built on by (most cited first, 1 of 1)

- [Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification](https://mlsanity.com/paper/2606.29440/) (2026)

## Abstract (arXiv)

Deep operator networks (DeepONets) represent a powerful class of data-driven methods for operator learning, demonstrating strong approximation capabilities for a wide range of linear and nonlinear operators. They have shown promising performance in learning operators that govern partial differential equations (PDEs), including diffusion-reaction systems and Burgers' equations. However, the accuracy of DeepONets is often constrained by computational limitations and optimization challenges inherent in training deep neural networks. Furthermore, the computational cost associated with training these networks is typically very high. To address these challenges, we leverage randomized neural networks (RaNNs), in which the parameters of the hidden layers remain fixed following random initialization. RaNNs compute the output layer parameters using the least-squares method, significantly reducing training time and mitigating optimization errors. In this work, we integrate DeepONets with RaNNs to propose RaNN-DeepONets, a hybrid architecture designed to balance accuracy and efficiency. Furthermore, to mitigate the need for extensive data preparation, we introduce the concept of physics-informed RaNN-DeepONets. Instead of relying on data generated through other time-consuming numerical methods, we incorporate PDE information directly into the training process. We evaluate the proposed model on three benchmark PDE problems: diffusion-reaction dynamics, Burgers' equation, and the Darcy flow problem. Through these tests, we assess its ability to learn nonlinear operators with varying input types. When compared to the standard DeepONet framework, RaNN-DeepONets achieves comparable accuracy while reducing computational costs by orders of magnitude. These results highlight the potential of RaNN-DeepONets as an efficient alternative for operator learning in PDE-based systems.