# A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains

arXiv: https://arxiv.org/abs/2409.05508 · HTML: https://mlsanity.com/paper/2409.05508/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 54 references that are in the map.
- First posted on arXiv 9 Sep 2024.

## Builds on (most cited first, 26 of 26)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Temporal Attention Unit: Towards Efficient Spatiotemporal Predictive Learning](https://mlsanity.com/paper/2206.12126/) (2022)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [OpenSTL: A Comprehensive Benchmark of Spatio-Temporal Predictive Learning](https://mlsanity.com/paper/2306.11249/) (2023)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [On the optimality of shape and data representation in the spectral domain](https://mlsanity.com/paper/1409.4349/) (2014)
- [Physics-Incorporated Convolutional Recurrent Neural Networks for Source Identification and Forecasting of Dynamical Systems](https://mlsanity.com/paper/2004.06243/) (2020)
- [LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data](https://mlsanity.com/paper/2206.09418/) (2022)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [Nonlinear model reduction for operator learning](https://mlsanity.com/paper/2403.18735/) (2024)
- [Transient Hemodynamics Prediction Using an Efficient Octree-Based Deep Learning Model](https://mlsanity.com/paper/2302.06557/) (2023)
- [Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processes](https://mlsanity.com/paper/2305.00094/) (2023)
- [Laplacian Spectral Basis Functions](https://mlsanity.com/paper/1906.03856/) (2019)

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

- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [A Physics-informed Multi-resolution Neural Operator](https://mlsanity.com/paper/2510.23810/) (2025)
- [Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data](https://mlsanity.com/paper/2511.18260/) (2025)

## Abstract (arXiv)

Predictive learning for spatio-temporal processes (PL-STP) on complex spatial domains plays a critical role in various scientific and engineering fields, with its essence being the construction of operators between infinite-dimensional function spaces. This paper focuses on the unequal-domain mappings in PL-STP and categorising them into increase-domain and decrease-domain mapping. Recent advances in deep learning have revealed the great potential of neural operators (NOs) to learn operators directly from observational data. However, existing NOs require input space and output space to be the same domain, which pose challenges in ensuring predictive accuracy and stability for unequal-domain mappings. To this end, this study presents a general reduced-order neural operator named Reduced-Order Neural Operator on Riemannian Manifolds (RO-NORM), which consists of two parts: the unequal-domain encoder/decoder and the same-domain approximator. Motivated by the variable separation in classical modal decomposition, the unequal-domain encoder/decoder uses the pre-computed bases to reformulate the spatio-temporal function as a sum of products between spatial (or temporal) bases and corresponding temporally (or spatially) distributed weight functions, thus the original unequal-domain mapping can be converted into a same-domain mapping. Consequently, the same-domain approximator NORM is applied to model the transformed mapping. The performance of our proposed method has been evaluated on six benchmark cases, including parametric PDEs, engineering and biomedical applications, and compared with four baseline algorithms: DeepONet, POD-DeepONet, PCA-Net, and vanilla NORM. The experimental results demonstrate the superiority of RO-NORM in prediction accuracy and training efficiency for PL-STP.