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.

翻译：深度神经算子（如DeepONets）已改变了高维非线性回归的范式，从函数回归转向（微分）算子回归，为计算工程应用的重大变革铺平了道路。本文研究了利用DeepONets推断未见过翼型周围流场以实现形状优化——这是空气动力学中一个通常消耗大量计算资源的重要设计问题。我们展示的结果表明，预测精度几乎无下降，同时在线优化成本降低了数个数量级。我们以NACA翼型作为所提方法的测试案例，因其形状可通过四位数参数化轻松定义。我们成功优化了带约束的NACA四位数问题，以最大化升阻比为目标，并通过与高阶CFD求解器对比验证所有结果。我们发现DeepONets具有低泛化误差，使其成为生成未见形状解的理想工具。具体而言，压力、密度和速度场可在不到一秒内精确推断，从而支持使用超越当前工作中所考虑的升阻比最大化的一般目标函数。