# Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems

arXiv: https://arxiv.org/abs/2204.09157 · HTML: https://mlsanity.com/paper/2204.09157/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 19 Apr 2022.

## Builds on (most cited first, 17 of 17)

- [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)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (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)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013)
- [Transfer Learning on Multi-Fidelity Data](https://mlsanity.com/paper/2105.00856/) (2021)
- [Neural Network Training Using ℓ1-Regularization and Bi-fidelity Data](https://mlsanity.com/paper/2105.13011/) (2021)
- [Generalized Joint Probability Density Function Formulation inTurbulent Combustion using DeepONet](https://mlsanity.com/paper/2104.01996/) (2021)

## Built on by (most cited first, 30 of 58)

- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Deep Operator Network Approximation Rates for Lipschitz Operators](https://mlsanity.com/paper/2307.09835/) (2023)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils](https://mlsanity.com/paper/2302.00807/) (2023)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [A DeepONet multi-fidelity approach for residual learning in reduced order modeling](https://mlsanity.com/paper/2302.12682/) (2023)
- [A multifidelity approach to continual learning for physical systems](https://mlsanity.com/paper/2304.03894/) (2023)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators](https://mlsanity.com/paper/2308.05141/) (2023)
- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Efficient kernel surrogates for neural network-based regression](https://mlsanity.com/paper/2310.18612/) (2023)
- [Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation](https://mlsanity.com/paper/2512.09165/) (2025)
- [Residual Multi-Fidelity Neural Network Computing](https://mlsanity.com/paper/2310.03572/) (2023)
- [Conformalized Prediction of Post-Fault Voltage Trajectories Using Pre-trained and Finetuned Attention-Driven Neural Operators](https://mlsanity.com/paper/2410.24162/) (2024)
- [Data-Efficient Deep Operator Network for Unsteady Flow: A Multi-Fidelity Approach with Physics-Guided Subsampling](https://mlsanity.com/paper/2503.17941/) (2025)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficients](https://mlsanity.com/paper/2311.03639/) (2023)
- [Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problems](https://mlsanity.com/paper/2401.07888/) (2024)
- [Branched Latent Neural Maps](https://mlsanity.com/paper/2308.02599/) (2023)
- [Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology](https://mlsanity.com/paper/2408.09818/) (2024)
- [Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources](https://mlsanity.com/paper/2508.19847/) (2025)
- [Mechanical Characterization and Inverse Design of Stochastic Architected Metamaterials Using Neural Operators](https://mlsanity.com/paper/2311.13812/) (2023)
- [Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs](https://mlsanity.com/paper/2502.00550/) (2025)
- [Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems](https://mlsanity.com/paper/2507.03660/) (2025)

## Abstract (arXiv)

Operator learning for complex nonlinear systems is increasingly common in modeling multi-physics and multi-scale systems. However, training such high-dimensional operators requires a large amount of expensive, high-fidelity data, either from experiments or simulations. In this work, we present a composite Deep Operator Network (DeepONet) for learning using two datasets with different levels of fidelity to accurately learn complex operators when sufficient high-fidelity data is not available. Additionally, we demonstrate that the presence of low-fidelity data can improve the predictions of physics-informed learning with DeepONets. We demonstrate the new multi-fidelity training in diverse examples, including modeling of the ice-sheet dynamics of the Humboldt glacier, Greenland, using two different fidelity models and also using the same physical model at two different resolutions.