# Temporal Neural Operator for Modeling Time-Dependent Physical Phenomena

arXiv: https://arxiv.org/abs/2504.20249 · HTML: https://mlsanity.com/paper/2504.20249/

## Key facts

- Cited by 8 papers in the map (data through 7 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 28 Apr 2025.

## Builds on (most cited first, 21 of 21)

- [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)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [Foundation Models for Weather and Climate Data Understanding: A Comprehensive Survey](https://mlsanity.com/paper/2312.03014/) (2023)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [State-space models are accurate and efficient neural operators for dynamical systems](https://mlsanity.com/paper/2409.03231/) (2024)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestration](https://mlsanity.com/paper/2409.16572/) (2024)
- [Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone](https://mlsanity.com/paper/2406.17119/) (2024)

## Built on by (most cited first, 8 of 8)

- [KANO: Kolmogorov-Arnold Neural Operator](https://mlsanity.com/paper/2509.16825/) (2025)
- [Autoregression-Free Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2605.25413/) (2026)
- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition](https://mlsanity.com/paper/2606.07601/) (2026)
- [Geometry-Aware Anisotropic Boundary Correction for Aerodynamic Simulation](https://mlsanity.com/paper/2606.09963/) (2026)
- [TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations](https://mlsanity.com/paper/2606.21189/) (2026)
- [Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets](https://mlsanity.com/paper/2608.13629/) (2026)
- [More Sensors Only One Field: Rethinking Continual Spatio-Temporal Forecasting](https://mlsanity.com/paper/2609.31325/) (2026)

## Abstract (arXiv)

Neural Operators (NOs) are machine learning models designed to solve partial differential equations (PDEs) by learning to map between function spaces. Neural Operators such as the Deep Operator Network (DeepONet) and the Fourier Neural Operator (FNO) have demonstrated excellent generalization properties when mapping between spatial function spaces. However, they struggle in mapping the temporal dynamics of time-dependent PDEs, especially for time steps not explicitly seen during training. This limits their temporal accuracy as they do not leverage these dynamics in the training process. In addition, most NOs tend to be prohibitively costly to train, especially for higher-dimensional PDEs. In this paper, we propose the Temporal Neural Operator (TNO), an efficient neural operator specifically designed for spatio-temporal operator learning for time-dependent PDEs. TNO achieves this by introducing a temporal-branch to the DeepONet framework, leveraging the best architectural design choices from several other NOs, and a combination of training strategies including Markov assumption, teacher forcing, temporal bundling, and the flexibility to condition the output on the current state or past states. Through extensive benchmarking and an ablation study on a diverse set of example problems we demonstrate the TNO long range temporal extrapolation capabilities, robustness to error accumulation, resolution invariance, and flexibility to handle multiple input functions.