# RONOM: Reduced-Order Neural Operator Modeling

arXiv: https://arxiv.org/abs/2507.12814 · HTML: https://mlsanity.com/paper/2507.12814/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 49 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 23 of 23)

- [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)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [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)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Discretization Error of Fourier Neural Operators](https://mlsanity.com/paper/2405.02221/) (2024)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [Uncertainty quantification of a three-dimensional in-stent restenosis model with surrogate modelling](https://mlsanity.com/paper/2111.06173/) (2021)
- [Joint Manifold Learning and Optimal Transport for Dynamic Imaging](https://mlsanity.com/paper/2505.11913/) (2025)

## Built on by (most cited first, 2 of 2)

- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)
- [Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2608.23546/) (2026)

## Abstract (arXiv)

Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification. Reduced-order modeling (ROM) addresses these challenges by constructing a low-dimensional surrogate model but relies on a fixed discretization, which limits flexibility across varying meshes during evaluation. Operator learning approaches, such as neural operators, offer an alternative by parameterizing mappings between infinite-dimensional function spaces, enabling adaptation to data across different resolutions. Whereas ROM provides rigorous numerical error estimates, neural operator learning largely focuses on discretization convergence and invariance without quantifying the error between the infinite-dimensional and the discretized operators. This work introduces the reduced-order neural operator modeling (RONOM) framework, which bridges concepts from ROM and operator learning. We establish a discretization error bound analogous to those in ROM, and get insights into RONOM's discretization convergence and discretization robustness. Moreover, three numerical examples are presented that compare RONOM to existing neural operators for solving partial differential equations. The results demonstrate that RONOM using standard vector-to-vector neural networks can achieve comparable performance in input generalization and achieves superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios and ROM-based approaches for learning on time-dependent data.