mlsanityData to 14 Oct 2026

Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport

arXiv 2204.06684 ↗First posted 14 Apr 2022physics.comp-ph cs.LGcited by 92

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Abstract (arXiv)

Deep neural operators can learn operators mapping between infinite-dimensional function spaces via deep neural networks and have become an emerging paradigm of scientific machine learning. However, training neural operators usually requires a large amount of high-fidelity data, which is often difficult to obtain in real engineering problems. Here, we address this challenge by using multifidelity learning, i.e., learning from multifidelity datasets. We develop a multifidelity neural operator based on a deep operator network (DeepONet). A multifidelity DeepONet includes two standard DeepONets coupled by residual learning and input augmentation. Multifidelity DeepONet significantly reduces the required amount of high-fidelity data and achieves one order of magnitude smaller error when using the same amount of high-fidelity data. We apply a multifidelity DeepONet to learn the phonon Boltzmann transport equation (BTE), a framework to compute nanoscale heat transport. By combining a trained multifidelity DeepONet with genetic algorithm or topology optimization, we demonstrate a fast solver for the inverse design of BTE problems.

Key facts. Cited by 92 papers in the map (data through 14 Oct 2026). Builds on 22 of its 57 references that are in the map. First posted on arXiv 14 Apr 2022.

Builds on

22 of 57 references.

References · 22

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  5. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  7. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  8. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  9. 2022MIONet: Learning multiple-input operators via tensor productcited by 181
  10. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  11. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  12. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
10 more
  1. 2022Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problemscited by 48
  2. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
  3. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  4. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  5. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  6. 2022DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectoriescited by 30
  7. 2022Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONetscited by 27
  8. 2022Systems Biology: Identifiability analysis and parameter identification via systems-biology informed neural networkscited by 23
  9. 2021Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEscited by 22
  10. 2019GMLS-Nets: A framework for learning from unstructured datacited by 18

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 12; 2023: 21; 2024: 20; 2025: 21; 2026: 18.

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  1. 2022Physics-Informed Deep Neural Operator Networkscited by 113
  2. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  3. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  4. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  5. 2023Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustnesscited by 60
  6. 2022Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problemscited by 58
  7. 2024Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolationcited by 46
  8. 2022SVD Perspectives for Augmenting DeepONet Flexibility and Interpretabilitycited by 35
  9. 2024RiemannONets: Interpretable Neural Operators for Riemann Problemscited by 28
  10. 2024Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networkscited by 28
  11. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  12. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
38 more
  1. 2023Improved generalization with deep neural operators for engineering systems: Path towards digital twincited by 19
  2. 2023Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoilscited by 19
  3. 2023A neural operator-based surrogate solver for free-form electromagnetic inverse designcited by 19
  4. 2023Learning in latent spaces improves the predictive accuracy of deep neural operatorscited by 19
  5. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 19
  6. 2022Multi-fidelity surrogate modeling using long short-term memory networkscited by 18
  7. 2023D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operatorscited by 18
  8. 2024Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solvingcited by 17
  9. 2023PROSE: Predicting Operators and Symbolic Expressions using Multimodal Transformerscited by 16
  10. 2024DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domainscited by 16
  11. 2023A DeepONet multi-fidelity approach for residual learning in reduced order modelingcited by 14
  12. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  13. 2024LeMON: Learning to Learn Multi-Operator Networkscited by 14
  14. 2023A multifidelity approach to continual learning for physical systemscited by 13
  15. 2023Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delaycited by 13
  16. 2022Multi-fidelity wavelet neural operator with application to uncertainty quantificationcited by 12
  17. 2023A Multifidelity deep operator network approach to closure for multiscale systemscited by 11
  18. 2024Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestrationcited by 11
  19. 2023Deep Operator Learning-based Surrogate Models with Uncertainty Quantification for Optimizing Internal Cooling Channel Rib Profilescited by 10
  20. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
  21. 2022Spiking Neural Operators for Scientific Machine Learningcited by 8
  22. 2024BiLO: Bilevel Local Operator Learning for PDE Inverse Problemscited by 8
  23. 2024Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problemcited by 8
  24. 2024Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneitycited by 8
  25. 2025TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansioncited by 8
  26. 2022Adaptive physics-informed neural operator for coarse-grained non-equilibrium flowscited by 7
  27. 2023Scattering with Neural Operatorscited by 7
  28. 2025Inverse Design in Nanophotonics via Representation Learningcited by 7
  29. 2025PI-MFM: Physics-informed multimodal foundation model for solving partial differential equationscited by 7
  30. 2023A hybrid Decoder-DeepONet operator regression framework for unaligned observation datacited by 6
  31. 2024Understanding the Difficulty of Solving Cauchy Problems with PINNscited by 6
  32. 2025Neural Emulator Superiority: When Machine Learning for PDEs Surpasses its Training Datacited by 6
  33. 2024Separable Operator Networkscited by 5
  34. 2025Data-Efficient Deep Operator Network for Unsteady Flow: A Multi-Fidelity Approach with Physics-Guided Subsamplingcited by 5
  35. 2025RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversioncited by 5
  36. 2023A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficientscited by 4
  37. 2024Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problemscited by 4
  38. 2024Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantificationcited by 4

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