Unlike physics-informed neural networks (PINNs) [Rai19P], a DeepONet [Lu21L] does not require any optimization during inference, hence it can be used in real-time forecasting. Traditional numerical models, such as compressible flow solvers, are computationally intensive for accurately modeling the flow field around complex airfoils. Surrogate models can alleviate the time-consuming optimization loop where the numerical solver calculates aerodynamic forces.

In a recent publication [Shu24D] in *Engineering
Applications of Artificial Intelligence*, the authors present a case study on
the use of DeepONet for airfoil shape optimization. They demonstrate
empirically that DeepONet can accurately predict flow fields around unseen
airfoils, cf. Figure 7, and serve as a fast surrogate for the optimization of
airfoil shapes with respect to a general objective function.

Specifically, the study optimizes the constrained NACA four-digit problem to maximize the lift-to-drag ratio. The results show minimal to no degradation in prediction accuracy using DeepONet while reducing the online optimization cost by 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 may
outweigh the benefits.

Remarkably, the authors investigate using a small dataset to train the surrogate
model: **40 training and 10 testing examples**. Yet, DeepONet generalizes well to
unseen airfoils (see Figure 9).

The paper effectively demonstrates an illustrative example within a general framework, showing how DeepONets can learn complex fluid dynamics and highlighting their potential to accelerate classical simulation tasks using deep learning.