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

arXiv: https://arxiv.org/abs/2301.06701 · HTML: https://mlsanity.com/paper/2301.06701/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 17 Jan 2023.

## Builds on (most cited first, 27 of 27)

- [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)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [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)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Discovering and forecasting extreme events via active learning in neural operators](https://mlsanity.com/paper/2204.02488/) (2022)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [Assessment of DeepONet for reliability analysis of stochastic nonlinear dynamical systems](https://mlsanity.com/paper/2201.13145/) (2022)
- [Fourier Neural Operator Networks: A Fast and General Solver for the Photoacoustic Wave Equation](https://mlsanity.com/paper/2108.09374/) (2021)
- [Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems](https://mlsanity.com/paper/2109.00538/) (2021)
- [Leveraging Industry 4.0 -- Deep Learning, Surrogate Model and Transfer Learning with Uncertainty Quantification Incorporated into Digital Twin for Nuclear System](https://mlsanity.com/paper/2210.00074/) (2022)
- [Digital Twin-Centered Hybrid Data-Driven Multi-Stage Deep Learning Framework for Enhanced Nuclear Reactor Power Prediction](https://mlsanity.com/paper/2211.13157/) (2022)
- [Structure and Distribution Metric for Quantifying the Quality of Uncertainty: Assessing Gaussian Processes, Deep Neural Nets, and Deep Neural Operators for Regression](https://mlsanity.com/paper/2203.04515/) (2022)
- [Machine learning based digital twin for stochastic nonlinear multi-degree of freedom dynamical system](https://mlsanity.com/paper/2103.15636/) (2021)
- [Data-driven multi-scale modeling and robust optimization of composite structure with uncertainty quantification](https://mlsanity.com/paper/2210.09055/) (2022)

## Built on by (most cited first, 19 of 19)

- [Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutions](https://mlsanity.com/paper/2308.07523/) (2023)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [Virtual Sensing-Enabled Digital Twin Framework for Real-Time Monitoring of Nuclear Systems Leveraging Deep Neural Operators](https://mlsanity.com/paper/2410.13762/) (2024)
- [Geometry-Informed Neural Operator Transformer](https://mlsanity.com/paper/2504.19452/) (2025)
- [An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Study](https://mlsanity.com/paper/2406.14715/) (2024)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [Real-Time Sensing of Inaccessible Physical Fields via an Edge-Deployable Hardware-Portable Graph Neural Operator](https://mlsanity.com/paper/2604.01802/) (2026)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [Agentic Physical AI toward a Domain-Specific Foundation Model for Energy Systems: A Case Study on Nuclear Reactor Control](https://mlsanity.com/paper/2512.23292/) (2025)
- [Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems](https://mlsanity.com/paper/2507.03660/) (2025)
- [Beyond Uniform Sampling: Synergistic Active Learning and Input Denoising for Robust Neural Operators](https://mlsanity.com/paper/2604.13316/) (2026)
- [Physics-informed Partitioned Coupled Neural Operator for Complex Networks](https://mlsanity.com/paper/2410.21025/) (2024)
- [Debiased neural operators for estimating functionals](https://mlsanity.com/paper/2604.19296/) (2026)
- [Sensing Without Colocation: Operator-Based Virtual Instrumentation for Domains Beyond Physical Reach](https://mlsanity.com/paper/2510.18041/) (2025)
- [Predicting Time-Dependent Flow Over Complex Geometries Using Operator Networks](https://mlsanity.com/paper/2512.04434/) (2025)
- [Disentangling Attention in Deep Operator Learning: A Controlled Study of Data-Driven and Physics-Informed Architectures](https://mlsanity.com/paper/2609.04407/) (2026)

## 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.