mlsanityData to 7 Oct 2026

Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils

arXiv 2302.00807 ↗First posted 2 Feb 2023physics.flu-dyn cs.AI math.OCcited by 19

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Abstract (arXiv)

Deep neural operators, such as DeepONets, have changed the paradigm in high-dimensional nonlinear regression from function regression to (differential) operator regression, paving the way for significant changes in computational engineering applications. Here, we investigate the use of DeepONets to infer flow fields around unseen airfoils with the aim of shape optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results which display little to no degradation in prediction accuracy, while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as their shape can be easily defined by the four-digit parametrization. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 11 of its 63 references that are in the map. First posted on arXiv 2 Feb 2023.

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11 of 63 references.

References · 11

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  2. 2015Automatic differentiation in machine learning: a surveycited by 846
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  5. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  6. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  7. 2022Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problemscited by 57
  8. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  9. 2022On the influence of over-parameterization in manifold based surrogates and deep neural operatorscited by 29
  10. 2022Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONetscited by 27
  11. 2020Enhanced data efficiency using deep neural networks and Gaussian processes for aerodynamic design optimizationcited by 15

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Citing papers per year: 2023: 8; 2024: 6; 2025: 3; 2026: 2.

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  1. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  2. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  3. 2024State-space models are accurate and efficient neural operators for dynamical systemscited by 27
  4. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  5. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 18
  6. 2024Diffeomorphism Neural Operator for various domains and parameters of partial differential equationscited by 15
  7. 2024DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domainscited by 14
  8. 2023Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operatorscited by 12
  9. 2023Interpreting and generalizing deep learning in physics-based problems with functional linear modelscited by 9
  10. 2023MyCrunchGPT: A chatGPT assisted framework for scientific machine learningcited by 8
  11. 2023A hybrid Decoder-DeepONet operator regression framework for unaligned observation datacited by 6
  12. 2023Grad-Shafranov equilibria via data-free physics informed neural networkscited by 5
7 more
  1. 2024Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformationscited by 5
  2. 2024Separable Operator Networkscited by 4
  3. 2025Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step casecited by 3
  4. 2025A Reward-Directed Diffusion Framework for Generative Design Optimizationcited by 2
  5. 2025DeepCFD: Efficient near-ground airfoil lift coefficient approximation with deep convolutional neural networkscited by 0
  6. 2026Optimization-Embedded Active Multi-Fidelity Surrogate Learning for Multi-Condition Airfoil Shape Optimizationcited by 0
  7. 2026Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solverscited by 0