Deep neural operators, such as DeepONet, have changed the paradigm in high-dimensional nonlinear regression, promising significant generalization and speed-up in computational engineering applications. In a recent paper, the authors investigate the use of DeepONet to infer flow fields around unseen airfoils with the aim of shape constrained optimization, an important design problem in aerodynamics that typically taxes computational resources heavily.Unlike physics-informed neural networks (PINNs) [Rai19P], a DeepONet [Lu21L] doesnot require any optimization during inference, hence it can be used in real-timeforecasting. Traditional numerical models, such as compressible flow solvers,are computationally intensive for accurately modeling the flow field aroundcomplex airfoils. Surrogate models can alleviate the time-consuming optimizationloop where the numerical solver calculates aerodynamic forces.In a recent publication [Shu24D] in EngineeringApplications of Artificial Intelligence, the authors present a case study onthe use of DeepONet for airfoil shape optimization. They demonstrateempirically that DeepONet can accurately predict flow fields around unseenairfoils, cf. Figure 7, and serve as a fast surrogate for the optimization ofairfoil shapes with respect to a general objective function. Figure 7: DeepONet Predictions. The pressure, density, andvelocity fields around the test set airfoil NACA 7315 predicted by the DeepONet,and the corresponding pointwise absolute errors are also provided.Specifically, the study optimizes the constrained NACA four-digit problem tomaximize the lift-to-drag ratio. The results show minimal to no degradation inprediction accuracy using DeepONet while reducing the online optimization costby approximately 30,000 times.How much data is needed?The crucial question is: how much data is needed to train the surrogate model?If the model requires too much data, the computational cost of training mayoutweigh the benefits.Remarkably, the authors investigate using a small dataset to train the surrogatemodel: 40 training and 10 testing examples. Yet, DeepONet generalizes well tounseen airfoils (see Figure 9).Figure 9: Plot of the computed lift-to-drag objective for theentire dataset sorted by the Nektar++ reference values. As seen in both plots,the approximation to the high-fidelity CFD solution is very accurate andconsistent throughout the entire parametric domain. Particularly, we note thatthe testing set performs comparably to the training set, meaning there is littleto no generalization error, which is necessary when inferring unseen queriedgeometries during optimization.The paper effectively demonstrates an illustrative example within a generalframework, showing how DeepONets can learn complex fluid dynamics andhighlighting their potential to accelerate classical simulation tasks using deeplearning.
