mlsanityData to 7 Oct 2026

Improved generalization with deep neural operators for engineering systems: Path towards digital twin

arXiv 2301.06701 ↗First posted 17 Jan 2023cs.LG stat.AP stat.COcited by 19

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Abstract (arXiv)

Neural Operator Networks (ONets) represent a novel advancement in machine learning algorithms, offering a robust and generalizable alternative for approximating partial differential equations (PDEs) solutions. Unlike traditional Neural Networks (NN), which directly approximate functions, ONets specialize in approximating mathematical operators, enhancing their efficacy in addressing complex PDEs. In this work, we evaluate the capabilities of Deep Operator Networks (DeepONets), an ONets implementation using a branch/trunk architecture. Three test cases are studied: a system of ODEs, a general diffusion system, and the convection/diffusion Burgers equation. It is demonstrated that DeepONets can accurately learn the solution operators, achieving prediction accuracy scores above 0.96 for the ODE and diffusion problems over the observed domain while achieving zero shot (without retraining) capability. More importantly, when evaluated on unseen scenarios (zero shot feature), the trained models exhibit excellent generalization ability. This underscores ONets vital niche for surrogate modeling and digital twin development across physical systems. While convection-diffusion poses a greater challenge, the results confirm the promise of ONets and motivate further enhancements to the DeepONet algorithm. This work represents an important step towards unlocking the potential of digital twins through robust and generalizable surrogates.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 27 of its 50 references that are in the map. First posted on arXiv 17 Jan 2023.

Builds on

27 of 50 references.

References · 27

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2014Deep Learning in Neural Networks: An Overviewcited by 765
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  5. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  6. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  7. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  8. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  9. 2018Deep Neural Networks Motivated by Partial Differential Equationscited by 187
  10. 2022Physics-Informed Deep Neural Operator Networkscited by 110
  11. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  12. 2021Improved architectures and training algorithms for deep operator networkscited by 85
15 more
  1. 2022Wavelet neural operator: a neural operator for parametric partial differential equationscited by 77
  2. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  3. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  4. 2022Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problemscited by 47
  5. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  6. 2022Discovering and forecasting extreme events via active learning in neural operatorscited by 32
  7. 2022Multi-fidelity wavelet neural operator with application to uncertainty quantificationcited by 12
  8. 2022Assessment of DeepONet for reliability analysis of stochastic nonlinear dynamical systemscited by 11
  9. 2021Fourier Neural Operator Networks: A Fast and General Solver for the Photoacoustic Wave Equationcited by 10
  10. 2021Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systemscited by 7
  11. 2022Leveraging Industry 4.0 -- Deep Learning, Surrogate Model and Transfer Learning with Uncertainty Quantification Incorporated into Digital Twin for Nuclear Systemcited by 7
  12. 2022Digital Twin-Centered Hybrid Data-Driven Multi-Stage Deep Learning Framework for Enhanced Nuclear Reactor Power Predictioncited by 6
  13. 2022Structure and Distribution Metric for Quantifying the Quality of Uncertainty: Assessing Gaussian Processes, Deep Neural Nets, and Deep Neural Operators for Regressioncited by 5
  14. 2021Machine learning based digital twin for stochastic nonlinear multi-degree of freedom dynamical systemcited by 4
  15. 2022Data-driven multi-scale modeling and robust optimization of composite structure with uncertainty quantificationcited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 1; 2024: 6; 2025: 8; 2026: 4.

19 papers.

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  1. 2023Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutionscited by 22
  2. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  3. 2024Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learningcited by 14
  4. 2025TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolationcited by 14
  5. 2024Virtual Sensing-Enabled Digital Twin Framework for Real-Time Monitoring of Nuclear Systems Leveraging Deep Neural Operatorscited by 12
  6. 2025Geometry-Informed Neural Operator Transformercited by 8
  7. 2024An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Studycited by 6
  8. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  9. 2025Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inferencecited by 5
  10. 2026Real-Time Sensing of Inaccessible Physical Fields via an Edge-Deployable Hardware-Portable Graph Neural Operatorcited by 4
  11. 2025Deep Neural ODE Operator Networks for PDEscited by 3
  12. 2025Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systemscited by 2
7 more
  1. 2024Physics-informed Partitioned Coupled Neural Operator for Complex Networkscited by 1
  2. 2025Agentic Physical AI toward a Domain-Specific Foundation Model for Energy Systems: A Case Study on Nuclear Reactor Controlcited by 1
  3. 2026Beyond Uniform Sampling: Synergistic Active Learning and Input Denoising for Robust Neural Operatorscited by 1
  4. 2026Debiased neural operators for estimating functionalscited by 1
  5. 2025Sensing Without Colocation: Operator-Based Virtual Instrumentation for Domains Beyond Physical Reachcited by 0
  6. 2025Predicting Time-Dependent Flow Over Complex Geometries Using Operator Networkscited by 0
  7. 2026Disentangling Attention in Deep Operator Learning: A Controlled Study of Data-Driven and Physics-Informed Architecturescited by 0