# ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines

arXiv: https://arxiv.org/abs/2501.09395 · HTML: https://mlsanity.com/paper/2501.09395/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 16 Jan 2025.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Variable-Input Deep Operator Networks](https://mlsanity.com/paper/2205.11404/) (2022)
- [The Deep Minimizing Movement Scheme](https://mlsanity.com/paper/2109.14851/) (2021)
- [Finite Element Operator Network for Solving Elliptic-type parametric PDEs](https://mlsanity.com/paper/2308.04690/) (2023)
- [Deep Neural Network Approach to Forward-Inverse Problems](https://mlsanity.com/paper/1907.12925/) (2019)
- [Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems](https://mlsanity.com/paper/2412.03161/) (2024)

## Built on by (most cited first, 3 of 3)

- [Hybrid Least Squares/Gradient Descent Methods for DeepONets](https://mlsanity.com/paper/2508.15394/) (2025)
- [Prescribed-Basis Coefficient-to-Coefficient Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2510.10350/) (2025)
- [Hybrid Least Squares/Gradient Descent Methods for MIONets](https://mlsanity.com/paper/2607.06976/) (2026)

## Abstract (arXiv)

Deep Operator Networks (DeepONets) are among the most prominent frameworks for operator learning, grounded in the universal approximation theorem for operators. However, training DeepONets typically requires significant computational resources. To address this limitation, we propose ELM-DeepONets, an Extreme Learning Machine (ELM) framework for DeepONets that leverages the backpropagation-free nature of ELM. By reformulating DeepONet training as a least-squares problem for newly introduced parameters, the ELM-DeepONet approach significantly reduces training complexity. Validation on benchmark problems, including nonlinear ODEs and PDEs, demonstrates that the proposed method not only achieves superior accuracy but also drastically reduces computational costs. This work offers a scalable and efficient alternative for operator learning in scientific computing.