Machine learning for adjoint vector in aerodynamic shape optimization

Abstract

Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for adjoint method to obtain the gradients of all design variables. However, the calculation cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the adjoint-based optimization efficiency, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction about NACA0012 airfoil. The results indicate that with negligible calculation cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.

Airfoil Analysis and Optimization Using a Petrov–Galerkin Finite Element and Machine Learning

“…Tyan et al [13] proposed an inverse design deep neural network (IDNN), dimensional reduction, and the NACA four-series airfoil geometry representation via 2D examples. In an alternative approach to optimization, Xu [14] proposed machine learning (ML) for the adjoint-variable method. In a parallel line of developments, it is worth mentioning the work by Gutierrez et al [15], which focused on the optimization of propellers for high-altitude pseudo-satellites.…”

AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems

“…Introduction. Optimal control modeling has been playing an important role in a wide range of applications, such as aeronautics [56], mechanical engineering [52], haemodynamics [43], microelectronics [39], reservoir simulations [57], and environmental sciences [48]. Particularly, to solve a PDE-constrained optimal control problem, one needs to find an optimal control function that can minimize a given cost functional for systems governed by partial differential equations (PDEs).…”

“…For example, a physics-informed neural network (PINN) method is designed to solve optimal control problems by adding the cost functional to the standard PINN loss [34,29]. Meanwhile, deep-learning-based surrogate models [56,30] and operator learning methods [55,20] are proposed to achieve fast inference for the optimal control solution without intensive computations. Although these methods are successful for solving optimal control problems, few of which can be directly applied in parametric optimal control modeling.…”