# Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations

arXiv: https://arxiv.org/abs/2212.04663 · HTML: https://mlsanity.com/paper/2212.04663/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 30 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [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)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [SURFNet: Super-resolution of Turbulent Flows with Transfer Learning using Small Datasets](https://mlsanity.com/paper/2108.07667/) (2021)

## Built on by (most cited first, 28 of 28)

- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Walrus: A Cross-Domain Foundation Model for Continuum Dynamics](https://mlsanity.com/paper/2511.15684/) (2025)
- [PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations](https://mlsanity.com/paper/2505.24717/) (2025)
- [PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Laws](https://mlsanity.com/paper/2401.07364/) (2024)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [Recurrent Neural Operators: Stable Long-Term PDE Prediction](https://mlsanity.com/paper/2505.20721/) (2025)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [MORPH: PDE Foundation Models with Arbitrary Data Modality](https://mlsanity.com/paper/2509.21670/) (2025)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [AMR-Transformer: Enabling Efficient Long-range Interaction for Complex Neural Fluid Simulation](https://mlsanity.com/paper/2503.10257/) (2025)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONets](https://mlsanity.com/paper/2306.15891/) (2023)
- [What do physics-informed DeepONets learn? Understanding and improving training for scientific computing applications](https://mlsanity.com/paper/2411.18459/) (2024)
- [Enhanced Vascular Flow Simulations in Aortic Aneurysm via Physics-Informed Neural Networks and Deep Operator Networks](https://mlsanity.com/paper/2503.17402/) (2025)
- [Implicit factorized transformer approach to fast prediction of turbulent channel flows](https://mlsanity.com/paper/2412.18840/) (2024)
- [Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge](https://mlsanity.com/paper/2602.15184/) (2026)
- [Long-time Integration of Nonlinear Wave Equations with Neural Operators](https://mlsanity.com/paper/2410.15617/) (2024)
- [Fourier analysis of the physics of transfer learning for data-driven subgrid-scale models of ocean turbulence](https://mlsanity.com/paper/2504.15487/) (2025)
- [A Physics-preserved Transfer Learning Method for Differential Equations](https://mlsanity.com/paper/2505.01281/) (2025)
- [Neural Operators for Multi-Task Control and Adaptation](https://mlsanity.com/paper/2604.03449/) (2026)
- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)
- [An optimal Petrov-Galerkin framework for operator networks](https://mlsanity.com/paper/2503.04024/) (2025)
- [Convolution Operator Network for Forward and Inverse Problems (FI-Conv): Application to Plasma Turbulence Simulations](https://mlsanity.com/paper/2602.04287/) (2026)
- [Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems](https://mlsanity.com/paper/2606.18305/) (2026)

## Abstract (arXiv)

Deep operator network (DeepONet) has demonstrated great success in various learning tasks, including learning solution operators of partial differential equations. In particular, it provides an efficient approach to predict the evolution equations in a finite time horizon. Nevertheless, the vanilla DeepONet suffers from the issue of stability degradation in the long-time prediction. This paper proposes a em transfer-learning aided DeepONet to enhance the stability. Our idea is to use transfer learning to sequentially update the DeepONets as the surrogates for propagators learned in different time frames. The evolving DeepONets can better track the varying complexities of the evolution equations, while only need to be updated by efficient training of a tiny fraction of the operator networks. Through systematic experiments, we show that the proposed method not only improves the long-time accuracy of DeepONet while maintaining similar computational cost but also substantially reduces the sample size of the training set.