# Multi-stage neural operator learning with application for convolutions

arXiv: https://arxiv.org/abs/2608.18851 · HTML: https://mlsanity.com/paper/2608.18851/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 50 references that are in the map.
- First posted on arXiv 19 Aug 2026.

## 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)
- [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)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)
- [Multi-Grade Deep Learning](https://mlsanity.com/paper/2302.00150/) (2023)
- [Unsupervised operator learning approach for dissipative equations via Onsager principle](https://mlsanity.com/paper/2508.07440/) (2025)
- [Machine-Learning Interatomic Potentials for Long-Range Systems](https://mlsanity.com/paper/2502.04668/) (2025)
- [A Fast Tree Algorithm for Electric Field Calculation in Electrical Discharge Simulations](https://mlsanity.com/paper/1710.05781/) (2017)

## Abstract (arXiv)

Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.